I found Akbar Ahmed's Discovering Islam to be a very useful one-volume introduction to the subject. It's particularly helpful for understanding Islam in South Asia.Depending on what kind of Islam or what part of the world you want to understand, of course, other books and authors would be better. For example, Ahmed Rashid's book on Central Asia is excellent.
To understand modern Iran, the satirical novel "My Uncle Napoleon" by Iraj Pezeshkzad and the two-volume graphic novel "Persepolis" by Marjane Satrapi are both excellent introductions. The first also became a popular TV series in Iran. If you want to work on a puzzle, watch the excellent 1960s-era Iranian film "Gaav" or "The Cow"--Khomeini loved this movie and watched it dozens of times in his life; the only movie he personally owned.
A very good non-fiction intro to Iran is called "Persian Postcards" by a Canadian journalist named Fred Reed. I don't know if it's available anymore.
As for the Arabic-speaking countries, Indonesia, or Turkey, I don't have much to add--I would have recommended Hourani's book as well.
[ 26 October 2007: Message edited by: rasmus ]