I am sure this question will have lots of answers as more people add more details. While not strictly taught in schools, I imagine you could fill an entire semester of college to answer just this one question.
To start. The official prayers in Judaism are a replacement of the sacrificial services done on the temple mount in Jeruslem. When the temple was destroyed, sacrifices were no longer capable of being given, and some reminder of those activities was needed. However, saying that, prayers are said to have existed since the time of Abrahahm, Issac and Jacob. Even when the temple was standing, people who could not make the pilgrimage would say prayers during the time of the sacrifices in lew of being able to be there.
All Jews prayer facing the Temple Mount in Jeruselem. That is where the old temple and Altar stood. However, if you are in a syngaggue, and there is a Torah scroll present, then one must pray facing the Torah Scroll. To alleviate confusion, most Torah scrolls are placed facing Jeruselem (but sometimes that isn't possible)
The text of the prayers differences from community to community, and wikipedia has an article on that here: http://en.wikipedia.org/wiki/Nusach
The differences between nusach are usually very minor. A few grammatical corrections here, a few extra or less words there. In general, a Jew visiting anywhere in the world can follow along with any prayer service. However, there are few Jewish groups who have made more than minor changes to the prayers. Some of these are accepted as ok universally, some are not.
There are two main types of prayers, known as the Shemah (Listen oh Israel), and the Amidah/Shemonah Esrei (The standing prayer, or the 18 blessings) The Shemah is said sitting down, and has 2 blessings before the shemah, and 2 blessings afterwards, while the Shemah itself consists of 3 paragraphs. More on that can be read about here: http://en.wikipedia.org/wiki/Amidah and here: http://en.wikipedia.org/wiki/Shema_Yisrael
There are other sections of prayer, depending on which service you are in. One such section are known as Psukei Dzimra (passages of song) which normally consist of various pslams from the Tanach (Hebrew scripture) Another common prayer is Ashrei ( http://en.wikipedia.org/wiki/Ashrei ) Also, there is a section of prayer which is a translation of the "Kedusha" into Aramaic. http://en.wikipedia.org/wiki/Kedushah
Lastly there is a prayer which is used to divide up the sections of prayers known as the Kaddish. http://en.wikipedia.org/wiki/Kaddish
Ok, not lastly.. there is also the prayer which is used to finish the services called Aleinu, ( http://en.wikipedia.org/wiki/Aleinu) and the song of the day which also comes from Psalms.
The Jewish prayer service has many different parts where a person must stand, some parts where a person must sit, and some parts which can be done in either way. In general, prayers that are 'more important', or where we are testifying our beliefs we stand. Also during the Amidah one must stand, and when the Torah is brought out. Different communities have different customs in this area.
Another point about prayer is that official Jewish Prayer normally breaks down into one of 3 categories. Thanks, Praise, or a Request. Anything that is not one of these categories is not 'officially' prayer.
I know I'm leaving something out, so I'll add more if there are follow up questions.