Observant Jews daven (pray) in formal worship services three times a day, every day: at evening (Ma'ariv), in the morning (Shacharit), and in the afternoon (Minchah). Daily prayers are collected in a book called a siddur, which derives from the Hebrew root meaning "order," because the siddur shows the order of prayers. It is the same root as the word seder, which refers to the Passover home service.
Undoubtedly the oldest fixed daily prayer in Judaism is the Shema. This consists of Deut. 6:4-9, Deut. 11:13-21, and Num. 15:37-41. Note that the first paragraph commands us to speak of these matters "when you retire and when you arise." From ancient times, this commandment was fulfilled by reciting the Shema twice a day: morning and night.
The next major development in Jewish prayer occurred during the Babylonian Exile, 6th century B.C.E. People were not able to sacrifice in the Temple at that time, so they used prayer as a substitute for sacrifice. "The offerings of our lips instead of bulls," as Hosea said. People got together to pray three times a day, corresponding to the three daily sacrifices. There was an additional prayer service on Sabbaths and certain holidays, to correspond to the additional sacrifices of those days. Some suggest that this may already have been a common practice among the pious before the Exile.
After the Exile, these daily prayer services continued. In the 5th century B.C.E., the Men of the Great Assembly composed a basic prayer, covering just about everything you could want to pray about. This is the Shemoneh Esrei, which means "18" and refers to the 18 blessings originally contained within the prayer. It is also referred to as the Amidah (standing, because we stand while we recite it), or Tefilah (prayer, as in The Prayer, because it is the essence of all Jewish prayer). This prayer is the cornerstone of every Jewish service.
The blessings of the Shemoneh Esrei can be broken down into 3 groups: three blessings praising G-d, thirteen making requests (forgiveness, redemption, health, prosperity, rain in its season, ingathering of exiles, etc.), and three expressing gratitude and taking leave. But wait! That's 19! And didn't I just say that this prayer is called 18?
One of the thirteen requests (the one against heretics) was added around the 2nd century C.E., in response to the growing threat of heresy (primarily Christianity), but at that time, the prayer was already commonly known as the Shemoneh Esrei, and the name stuck, even though there were now 19 blessings.
Another important part of certain prayer services is a reading from the Torah (first 5 books of the Bible) and the Prophets. The Torah has been divided into 54 sections, so that if each of these sections is read and studied for a week, we can cover the entire Torah in a year every year (our leap years are 54 weeks long; regular years are 50 or so, we double up shorter portions on a few weeks in regular years). At various times in our history, our oppressors did not permit us to have public readings of the Torah, so we read a roughly corresponding section from the Prophets (referred to as a Haftarah). Today, we read both the Torah portion and the Haftarah portion. These are read on Mondays, Thursdays, Sabbaths and some holidays. The Torah and haftarah readings are performed with great ceremony: the Torah is paraded around the room before it is brought to rest on the bimah (podium), and it is considered an honor to have the opportunity to recite a blessing over the reading (this honor is called an aliyah).
That's the heart of the Jewish prayer service. There are a few other matters that should be mentioned, though. There is a long series of morning blessings at the beginning of the morning service. Some people recite these at home. They deal with a lot of concerns with getting up in the morning, and things we are obligated to do daily. There is a section called Pesukei d'Zemira (verses of song), which includes a lot of Psalms and hymns. I like to think of it as a warm-up, getting you in the mood for prayer in the morning.
There are also a few particularly significant prayers. The most important is the Kaddish, the only prayer in Aramaic to my knowledge, which praises G-d. Here's a small piece of it, in English:
There are several variations on it for different times in the service. One variation is set aside for mourners to recite, the congregation only providing the required responses. Many people think of the Kaddish as a mourner's prayer, because the oldest son is obligated to recite it for a certain period after a parent's death, but in fact it is much broader than that. Someone once told me it separates each portion of the service, and a quick glance at any siddur (daily prayerbook) shows that it is recited between each section, but I don't know if that is its purpose.
Another important prayer is Aleinu, which is recited at or near the end of every service. It also praises G-d. Here is a little of it in English, to give you an idea:
On certain holidays, we also recite Hallel, which consists of Psalms 113-118.
Many holidays have special additions to the liturgy. See Yom Kippur Liturgy for additions related to that holiday.
There are a few other things, but that's a pretty good idea of what's involved. Here is an outline of the order of the daily services:
This is based on the Ashkenazic service, but the Sephardic service has a very similar structure. They use different music, and have a few variations in choice of psalms, hymns, and prayers.
The above is from the Orthodox prayerbook. The Reform service, although much shorter, follows the same basic structure and contains shorter versions of the same prayers with a few significant changes in content (for example, in one blessing of the Shemoneh Esrei, instead of praising G-d who "gives life to the dead," they praise G-d who "gives life to all" because they don't believe in resurrection). The Conservative version is very similar to the Orthodox version, and contains only minor variations in the content of the prayers (similar to the Reform example).
There are a few significant differences in the way that services are conducted in different movements:
Sources: Judaism 101; Much of the information in this page is derived from Rabbi Hayim Halevy Donin's "To Pray as a Jew: A Guide to the Prayer Book and the Synagogue Service", an excellent Orthodox resource on the subject of Jewish prayer.