Islamic history is not my specialty, so perhaps someone more knowledgeable can give you more, but here's what I know. The theology is a little weak for me (not being Muslim), though I spoke with Muslim students on the subject during the Jyllands-Posten controversy. Not useful for a strict theological interpretation, but useful for understanding what the "common man" believes. I'm also not sure whether or not they would qualify as a reliable source for that reason so I hope I'm not stepping on any toes or anything.

First a little background. The pre-Islamic Arab religions were polytheistic and associated with idols and idolatry. We can tell this various ways, and the Book of Idols by Kitab al Asnam mentions a number of deities and myths. Islam is very strongly against Idolatry (shirk). They consider it as worshiping an object in place of Allah, or worshiping the material world to Allah's exclusion. Both inherently deny Allah's power and majesty, and thus constitute a rejection of Him (riddah). It is an unforgivable sin if one does not renounce it before death. It's a very big deal.

Here's where the Muslim students' opinions come in: That said, artistic depictions of the world are seen similarly - as mere facsimiles of the real world, without any of the power or sanctity. Representational art can be seen, depending on theological school of thought, as a crude and imprecise attempt to replicate Allah's creation, making a mockery of the sophistication of the world for one's viewing pleasure. Attempting to cover up the realities of the world and substitute one's own, with an implication that it's somehow "better". Similarly, depicting Mohammad in any way will inevitably not be him, or look like him, it will be a crude misrepresentation - an idol (as a side note, this is also the theological reasoning behind the Saudi destruction of early Islamic sites like the house of Mawlid where Muhammad was believed to have been born, and Jannatul Baqi; the Saudis believe the sites themselves are being worshiped or distracting from the worship of Allah). Therefore it's disrespectful and religiously dangerous to try and "substitute" the real Muhammad with a fanciful graphical depiction. Television and photography among some Islamic schools of thought are also to be considered taboo.

That's why calligraphy is so frequently used as a substitute for portraits - and those particularly good at calligraphy even go so far as to write out entire verses or pages of a book in the shape of an animal. In illuminated manuscripts, you'll occasionally see miniatures that include depictions of soldiers, hunters, or notable councils as well.

Now with that background of where it seems to come from, if we look back historically we know this was not always so strict. Persia for example has a strong, relatively unbroken representational tradition influenced in particular by Chinese art. The Ottomans and Mughals both had strong representational traditions as well, largely influenced in turn by the Persian style. We also know that the coins of the early Islamic period, up until about the time of the Caliph Abd Al-Malik ibn Marwan (about 650-700AD), had images of people (frequently the Caliph, in copying the Byzantine pattern) and Islamic symbolism on each side. After Abd Al-Malik, coins only had writing struck on them in most of the Arab world. Fringe areas like the Seljuk empire continued to use representational art on their coins.

Interesting to note, modern Islamic currencies depict representational art, in many cases containing very realistic portraits of their leaders (Iranian rials have the Ayatollah Khomeini, Saudi Arabia has Abdulaziz, etc) so the proscription has clearly changed with regards to certain subjects.

When exactly the prohibition occured I don't believe we know for sure, but it seems to have happened quite early in Islamic history (in fact, perhaps during Abd Al-Malik's reign), but was essentially localized to the core Islamic area of the Arabian Peninsula up through to Syria and the Levant. Egypt never seems to have completely done away with representational art, as we have occasional metalwork and lacquer-ware with human and animal figures, especially from the Mamluk period. I'm unsure about any possible extent of representational art in North Africa and Islamic Spain, but they seem to have very strongly preferred geometric art.

A good book on idolatry in Islam, if you're curious, is The idea of idolatry and the emergence of Islam : from polemic to history by G.R. Hawting. Chapter 3 deals with idolatry in Islam, which is fundamentally what the issue revolves around.

May I ask a follow up question: was there any connection between Islamic views on the depiction of Mohammad and Protestant iconoclasm during the Reformation? Would Martin Luther's and John Calvin's views have been influenced by the Islamic world, either directly or via Eastern Christianity?