The amateur radio service has its own designated format for call signs: one or two letters, a single digit, then one to three letters. These days, the FCC normally issues a new licensee a "two by three" call sign (two-letter prefix, three-letter suffix), but depending on your license class, you can qualify for a shorter one. The Advanced and Amateur Extra license classes are allowed to have 1x2 or 2x1 call signs. According to the ARRL[1], there are currently 38093 Advanced licensees and 150793 Extra licensees (Advanced is no longer issued, but there are still quite a few Advanced who never upgraded to Extra).
Of course, a call sign can only be held by one licensee at a time, and most (or all) of the 1x2 and 2x1 are assigned (I'm not sure on the current state of things with 1x3 and 2x2). So one only becomes available when the previous holder loses it (by license expiration or death). There are far more Advanced/Extra licensees than short call signs.
There are also 1x1 call signs, but they are only issued on a temporary basis to special event stations (e.g., I have operated W8C, with a club whose normal call sign is W8YY).
Obviously, getting a randomly assigned call sign that happens to match your initials would be pretty rare. However, the FCC allows licensed holders who hold a General or above, to apply for a "vanity license" whereby a request is made for an unassigned or abandoned call sign on a first come, first serve basis. Abandoned call signs are those where the owner has either died or has not renewed their license for a period of 3 years following the expiration of the license. This is how I picked up my current call sign and I'm pretty sure that Joe Taylor got his this way.
To add to this how non-US licenses are done, there is an additional rule with a number prefix, For example a station in Kuwait might have a call beginning with 9K2.
Additionally, some have two digits, for example a Slovenia call might be S52.
So the above rule would be modified to be zero or one digit, one or two letters, and one or two numbers, followed by suffix.
Of course, a call sign can only be held by one licensee at a time, and most (or all) of the 1x2 and 2x1 are assigned (I'm not sure on the current state of things with 1x3 and 2x2). So one only becomes available when the previous holder loses it (by license expiration or death). There are far more Advanced/Extra licensees than short call signs.
There are also 1x1 call signs, but they are only issued on a temporary basis to special event stations (e.g., I have operated W8C, with a club whose normal call sign is W8YY).
1: http://www.arrl.org/fcc-license-counts