I always hear the terms "apicorsis" and "kofer״, along with "kefirah" and amaratzit. What is the definition of each?
"Am Haaretz" just means "ignoramus" colloquially but the rest are defined by Rambam in Hilchos Teshuvah Chapter 3.
Rambam identifies four kinds of heretics (this is a subset of those who "have no share in the world to come").
Three kinds of kofer: One who denies the divine origin of any portion of the written Torah; one who denies the validity of the oral Torah or the authenticity of it's teachers; and one who says that the mitzvos or any given mitzvah has been replaced with something else.
Five kinds of min: Atheists or deists (one kind); polytheists; corporealists; those who deny creation ex nihilo; and those who worship intermediaries.(Min comes from the word for "species" or "kind" and might refer to sectarians.)
Three kinds of Apikoyris: One who denies the existence of prophecy; one who denies all or some of Moshe's prophecy; and one who denies the idea that God knows what people are doing.
"Apikorus" derives from the Greek Eπικουρος (epikouros) - the name of a philosopher ("Epicurus" in Latin) who believed, among other things, that the gods had abandoned this world after having created it. An epicurean, in this context, is one who rejects any belief in divine providence, God's involvement in human history and in revelation. According to the Mishna (Sanhedrin 10:1), such a person does not qualify for the world to come.
"Kofer" derives from the Hebrew כופר ("one who covers up") and refers to one who actively denies something that they should otherwise know. In practise, it refers to one who denies the existence of God altogether. "Kefirah" is an abstract noun formed off the same root ("the act/nature of covering up").
"Amaratzit", as you spelt it in your question, is an abstract noun formed from the phrase Am haAretz (עם הארץ), which means "people of the land". Originally a plural designation, it has come over time to function also in a singular capacity. It is attested several times within Tanakh, but takes on a substantially more negative nuance in the rabbinic literature. Some passages that speak of the Am haAretz are decidedly harsher than others, but today it has more of the sense of "ignoramus", rather than anything malicious.
Here's something on "apikoros"
Tov Halevanon commentary on shaar yichud ch.2 of Chovos Halevavos: