Services are composed of Bible readings, Temple symbolism, and public and private prayer. The liturgy itself combines biblical passages with poetry and prose by later writers.
The core prayers of the service are the Shema and the Amidah.
The Shema consists of three paragraphs from the Pentateuch (Deut. 6:4-9, Deut. 11:13-21, Num. 15:37-41). The Shema is always read evening and morning because of the words found in the first two paragraphs, which admonish us to speak these words when "you lie down and when you rise up."
The Amidah represents the ancient Temple sacrifices. The priests used to offer a sacrifice every morning (the Shaharit). After the destruction of the Temple by the Romans, the rabbis ordained that prayer should be substituted for sacrifice and the tradition arose of reciting a special prayer while standing. This prayer came to be known as the Amidah.
On the Sabbath there is also a public reading from the Torah scroll.
Preliminary benedictions and psalms were added over time because there was a feeling that one needed some preparation to get into the proper mood for prayer, and some final benedictions and hymns were also added to create more of a sense of closure to the service.
A recitation known as the Kaddish came to be added as punctuation, marking the various divisions of the service. (The words Kaddish, Kedushah, and Kiddush are all variants on the Hebrew term usually translated as "holy.") The final Kaddish in a service is designated as the Mourners' Kaddish.
There is also a more detailed guide to the specific parts of the Shabbat morning service at Shir Hadash.