The Dome of the Rock (Qubbat As-Sakhrah in Arabic), is one of the most important holy sites of Islam, and is the first example of really massive and permanent Islamic monumental architecture. Though commonly conceived of by westerners as a mosque, it isn't one; a mosque is a site for communal worship, and the dome is a commemorative shrine (masshad), built on a holy site, in this case the top of the Temple Mount. Even Muslims misconceive the purpose of its construction, as most would tell you that it commemorates the site of The Prophet's Night Journey, his bodily visit to heaven - but the identification of the peak of Mt. Moriah as the site of the Night Journey wasn't made until about the 10th century CE, and the Dome was completed in 691.
The Umayyad caliph Abd al-Malik commissioned the Dome for reasons that went unrecorded, but that we can speculate on. There would have been four important things that made the peak of the Temple Mount important in the Muslim imagination in the 7th century: that it was site that Adam fell to earth from after his expulsion from heaven (Adam's fall is considered to be both literal and metaphorical in the Islamic tradition), that it was the spot where Abraham was ordered by God to sacrifice his son, that it was the site of the First and Second Temples in Judaism, and that it was the center of the first qibla, or direction of prayer, before it was changed to Mecca. It's worth noting that Islamic tradition treats Abraham's sacrifice quite a bit differently, and more ambiguously, than Jewish and Christian sources. The Qur'an never mentions whether the son that Abraham was ordered to sacrifice was Isaac, the ancestor of the Jews, or Ishmael, ancestor of the Arabs, and traditions exist within Islam for both. Also, in some Muslim readings of the sacrifice, Abraham actually did sacrifice his son, who was then resurrected by God. Nowadays, most Muslims would place the sacrifice as having occured in Saudi Arabia, near Mecca, rather than on the Temple Mount as was the prevalent view at the time.
Another fact that would certainly be on Abd al-Malik's mind is that his capital was in Damascus, just north of Jerusalem and far from Mecca, and that Mecca had recently been the site of an abortive revolution against Umayyad rule. Solidifying the symbolic importance of Jerusalem would have been an important goal for him, and the most noticeable architectural feature of Jerusalem up to that point would have been the Byzantine Christian Church of the Holy Sepulchre. While Jerusalem had been under Byzantine domination, they had done everything in their power to solidify Christianity's hold on Jerusalem - in addition to building the Holy Sepulchre, they had expelled all Jews from the city, and turned the Temple Mount into a garbage dump. Islam was in much closer alliance with Judaism than Christianity during its earlier period, and two of Umar's first actions on taking Jerusalem had been to re-admit Jews into the city and to order the clearing of the garbage which had accumulated on the Temple Mount.
The Dome of the Rock then, was a building built with many political and religious goals, primarily to establish both Islamic and Umayyad claims on Jerusalem, to serve as a sort of architectural rebuke to the Church of the Holy Sepulchre, and ironically, to bring Islam in closer harmony with the older Monotheistic traditions of Judaism. The dome itself is a sixty foot high vaulted dome (slightly higher than the dome of the Byzantine Holy Sepulchre). It is currently covered on the outside with gold leaf, a gift of the Jordanian government, but was probably originally faced with polished glass mosaic-work.
During the century that the Crusaders occupied Jerusalem they believed that the Dome was actually the first, or Solomon's temple, which had stood on the same site a few thousand years earlier (they also believed that the al-Aqsa mosque was Solomon's palace). This is probably why it wasn't destroyed and replaced with a church. The Templum Deum became one of the major landmarks of Crusader Jerusalem - it's the temple in the name of the Knights Templar. Ironically, some of the Arabic inscriptions on the inside walls of the Dome are quotes from the Qur'an that specifically deny the Doctrine of the Trinity and the divinity of Christ.
Presently, the Dome is the site of major contention. Many of the more radical ultra-Orthodox Jewish groups in Israel propose razing the dome so that the Temple can be rebuilt - a position that the Israeli government has repeatedly rebuked for the obvious reason that it would start a major war.