The Rambam explains many of the fundamentals of Judaism in his Mishneh Torah. You can find many of the concepts in Hilchot Yesodei Hatorah and Hilchot Teshuvah.
The Rambam also outlines thirteen fundamental principles of Judaism. You can see a translation of the principals by Marc Mermelstein below:
The 13 Foundations of Judaism
Principle I. To know the existence of the Creator
To believe in the existence of the Creator, and this Creator is perfect in all manner of existence. He is the cause of all existence. He causes them to exist and they exist only because of Him. And if you could contemplate a case, such that He was not to exist…then all things would cease to exist and there would remain nothing. And if you were to contemplate a case, such that all things would cease to exist aside from the Creator, His existence would not cease. And He would lose nothing; and oneness and kingship is His alone. Hashem of strength is His name because He is sufficient with His own existence, and sufficient [is] just Him alone, and needs no other. And the existences of the angels, and the celestial bodies, and all that is in them and that which is below them…all need Him for their existence. And this is the first pillar and is attested to by the verse, “I am Hashem your God.” (Elaboration on this principle)
Principle II. The unity of God
Meaning to say to accept that this is the quintessential idea of Oneness. It is not like the oneness of a pair (i.e. pair of shoes - one group) and not one like a species. And not like man that has many individual (members) nor like a body that divides into many different parts until no end (every part being divisible). Rather, God is one and there is no other oneness like His. This is the second principle and is taught in what it says, “Hear Israel, Hashem is Our God, Hashem is one.”
Principle III. The denial of physicality in connection with God
This is to accept that this Oneness that we have mentioned above (Principle II) is not a body and has no strength in the body, and has no shape or image or relationship to a body or parts thereof. This is why the Sages of blessed memory said with regards to heaven there is no sitting, nor standing, no awakeness, nor tiredness. This is all to say that He does not partake of any physical actions or qualities. And if He were to be a body then He would be like any other body and would not be God. And all that is written in the holy books regarding descriptions of God, they are all anthropomorphic. Thus said our great Rabbis of blessed memory, “The Torah speaketh in man’s language” (i.e. using human terms to offer some understanding). And the Rabbis have already spoken at length on this issue. This is the third pillar and is attested to by the verse, “For you saw no image” meaning that you did not see an image or any form when you stood at Sinai because as we have just said, He has no body, nor power of the body.
Principle IV. God’s Antiquity
This is that God existed prior to everything, and exists after everything. This is proved many times throughout scripture and is attested to by the verse, “Meuna Elokei kedem.”
Principle V. That God, blessed be He is worthy that we serve Him, to glorify Him, to make known His greatness, and to do His commands
But not to do this to those that are below Him in the creation. Not to the angels or to the stars or the planets or anything else, for they are all created things in nature and in their functioning, there is no choice or judgment except by God Himself. Also it is not fitting to serve them as intermediaries to God. Only to God should you incline your thoughts and your actions. This is the fifth principle and it warns against idolatry and most of the Torah speaks out against this.
Principle VI. Prophecy
And this is that it is known to man that this (prophet) is a type of man who are created beings of great stature and perfection of the character traits. Who have tremendous knowledge until a different intelligence attaches to them when the intelligence of the person clings to the intelligence of God and it rests upon him. And these are the prophets; and this is prophecy; and the idea of it. The explanation of it is very long and the intention is not to bring a sign for every fundamental and to explain it all, encompassing of all knowledge (i.e. God’s knowledge) but it is mentioned to us in a story form and all of the Torah attests to this.
Principle VII. The prophetic capacity of Moses our Teacher, peace be upon him
And this is that we accept that he was the father of all prophets that were before him and that will be after him. He was on a qualitatively different level than any other, and he is chosen from all other people before and after him of any that have any knowledge of God; for his was the greatest. And he, peace be upon him, rose to the levels of the angels. He was granted all areas of knowledge and prophecy and his physical attributes did not diminish. His knowledge was different and it is through this difference that it is ascribed to him that he spoke to God without any intermediary or angel.
My intention was to explain this puzzling concept and to open up the sealed areas in the Torah regarding the verses of “face to face” and other similar references, but its length would be tremendous and it would require numerous proofs from the Torah and other sources and encompass many areas. Even to write it the briefest of briefest it would require 100 pages, so I will save it and write it in another book. I will now return to the intent of this seventh fundamental that the prophecy of Moses our teacher, peace be upon him, was different from all others in 4 ways:
1) Regarding all other prophets, God spoke to them through intermediaries. Regarding Moses, it was without one, as it says, “face to face I spoke to him”.
2) Regarding all other prophets, prophecy came to them at night while they were asleep in a dream as it says, “in a dream of the night” and other such references; or in the day but only after a deep sleep-like state came over them, and all their senses were shut off except their thoughts. Not so by Moses. Moses would receive a prophecy any time when he would stand between the two figures [fixed] on the ark, as God attests to it, “and I will make it known to you there” and “not so my servant Moses. Face to face I speak to him.”
3) When a prophet would receive prophecy he would not be able to stand the intense effect and he would shake and not be able to stand, as it relates regarding Daniel in his encounter with the angel Gabriel. Regarding Moses, he did not suffer from this. As it says, “Face to face do I speak to him as a person speaks to his friend”. And even though this is the greatest connection to God, still, he did not suffer.
4) All other prophets could not receive prophecy at their will, [but] only when God desired to tell them. Some would go days or months without prophecy. Even if they wanted or needed something, sometimes it would be days or months or years or even never that they would be told [a prophecy]. Some would have people play music to put them in a good mood such as Elisha. But Moses, peace be upon him, received prophecy whenever he wanted, as it says, “Stand here and listen to what God will tell you what to do” and “God said to Moses tell Aaron your brother that he can’t come to the holy of holies at any time [he wants]”. Our rabbis said, “Aaron was prohibited to come whenever he wanted, but not Moses.
Principle VIII. That the Torah is from heaven [God]
And this is that you believe that all of this Torah that was given by Moses our teacher, peace be upon him, that it is all from the mouth of God. Meaning that it was received by him entirely from God. And it is not known how Moses received it except by Moses himself, peace be upon him, that it came to him. That he was like a stenographer that you read to him and he writes all that is told to him: all the events and dates, the stories, and all the commandments. There is no difference between “And the sons of Cham were Kush, and Mitzraim, and his wife was Mehatbe’el” and “Timnah was his concubine” and “I am Hashem your God” and “Hear Israel [Hashem your God, Hashem is one]” for it was all given by God. And it is all Hashem’s perfect Torah; pure, holy, and true. And he who says that these verses or stories, Moses made them up, he is a denier of our sages and prophets worse than all other types of deniers [form of heretic] for he thinks that what is in the Torah is from man’s flawed heart and the questions and statements and the dates and stories are of no value for they are from Moses Rabbeinu, peace be upon him. And this area is that he believes the Torah is not from heaven. And on this our sages of blessed memory said, “he who believes that the Torah is from heaven except this verse that God did not say it but rather Moses himself did [he is a denier of all the Torah].” And this that God spoke this and that, each and every statement in the Torah, is from God and it is full of wisdom (each statement) and benefit to those who understand them. And its depth of knowledge is greater than all of the land and wider than all the seas and a person can only go in the path of David, the anointed of the God of Jacob who prayed and said “Open my eyes so that I may glance upon the wonders of Your Torah” (Psalms 119). And similarly the explanation of the Torah was also received from God and this is what we use today to know the appearance and structure of the sukka and the lulav and the shofar, tzitzis, tefillin and their usage. And all this God said to Moses and Moses told to us. And he is trustworthy in his role as the messenger and the verse that teaches of this fundamental is what is written (Numbers 16) “And Moses said, with this shall you know that Hashem sent me to do all these actions (wonders) for they are not from my heart.”
Principle IX. The completeness of the Torah
And this is that the Torah is from God and is not lacking. That to it you can’t add or take away from. Not from the written Torah or from the oral Torah, as it says, “Do not add to it and do not take away from it.” (Deut 13:1). And we already explained what needs to be explained about this fundamental at the beginning of this essay.
Principle X. That God knows man’s actions and does not remove His eye from them
His knowledge is not like someone who says God abandoned the land but rather like it says (Jer. 32) “Great in council and mighty in deed, Your eyes are cognizant to all the ways of mankind.” “And God saw for the evil of man on the land had grown greatly.” (Gen. 6) And it says, “The disgust of Sodom and Amorrah is great” and this demonstrates the 10th principle.
Principle XI. That God gives reward to he who does the commandments of the Torah and punishes those that transgress its admonishments and warnings
And the great reward is the life of the world to come and the punishment is the cutting off of the soul [in the world to come]. And we already said regarding this topic what these are. And the verse that attests to this principle is (Exodus 32) “And now if You would but forgive their sins - and if not erase me from this book that You have written.” And God answered him, “He who sinned against Me I will erase from My book.” This is a proof that God knows the sinner and the fulfiller in order to mete out reward to one and punishment to the other.
Principle XII. The era of the Messiah
And this is to believe that in truth that he will come and that you should be waiting for him even though he delays in coming. And you should not calculate times for him to come, or to look in the verses of Tanach to see when he should come. The sages say: The wisdom of those who calculate times [of his coming] is small and that you should believe that he will be greater and more honored than all of the kings of Israel since the beginning of time as it is prophesied by all the prophets from Moses our teacher, peace be upon him, until Malachi, peace be upon him. And he who doubts or diminishes the greatness of the Messiah is a denier in all the Torah for it testifies to the Messiah explicitly in the portion of Bilam and the portion of “You are gathered (towards the end of Deut)”. And part of this principle that there is no king of Israel except from the house of David and from the seed of Solomon alone. And anyone who disputes this regarding this family is a denier of the name of God and in all the words of the prophets.
Principle XIII. Resurrection of the dead
And we have already explained it And when the person will believe all these fundamentals and his faith will be clear in them he enters into the nation of Israel and it is a mitzva to love him and to have mercy on him and to act to him according to all the ways in which God commanded us regarding loving your neighbor. And even if he did all of the sins in the Torah due to desire of the emotions, and from his physical aspect’s conquering him, he will be punished for his sins, but he still has a share in the world to come and is among the sinners of Israel. However if he rejects one of these fundamentals he leaves the nation and is a denier of the fundamentals and is called a heretic, a denier, etc., and it is a mitzva to hate him and to destroy him (financially - not physically to kill him. And not to steal either). And regarding him it is said (Psalms 139) “Behold will not the enemy of God be my enemy?”
I have expounded at length many things and I have left the topic of my composition but I have done it for I saw a need in the dealings of the fundamentals of faith and I have gathered together many different and spread out areas Therefore know them and succeed in understanding them and review them many times and know them very well [i.e. not just memorization but to understand fully and to be able to support them and know their proofs]. Therefore if after one or ten times you think you have understood them, God knows that you are just involved in falsehood. Therefore do not read them quickly because I have not written them as it suddenly entered into my mind. But rather, after a deep and careful study of the whole area and after I have seen many clear and true ideas and I have seen what is proper to believe of them [as the fundamentals] and I have brought proofs and logical demonstrations for each and every one of them. May it be God’s will that I have been correct that He helped me through this area on the good path and now I will return to my explanation of this chapter [in the Talmud].
(To read the actual writing of Maimonides in Hebrew, you may locate the Principles at the end of the Talmud Sanhedrin, in Maimonides' commentary on the Mishna.)