Edited February 2004 to be less ugly and generally clunky. Ye gods.
The borough is the county-level government structure in Alaska. However, unlike Louisiana's parishes (which are counties in all but name), the Alaska borough is a somewhat different animal. A borough is governed by an assembly, which is identical to a city council, only with a slightly cooler name. There are two main types of boroughs (and it should go without saying that outside of lawyers, nobody actually uses these terms):
Home rule boroughs are those that have adopted a home rule charter and have "all legislative powers not prohibited by law or charter". In English, this means they can do anything the state hasn't poached for itself.
A subset of these are unified municipalities, which are weird hybrids of city and county. (Indianapolis's Unigov is, from what little I know about it, the same idea as this) They occur when a borough and everything inside it merges into a single unit. Essentially it's the city annexing the entire borough, though tbe borough assembly structure is kept.
General law boroughs have only the powers given them under the Alaska Statutes. These are given according to the borough's class, of which there are three, imaginatively called first, second, and third class:
Third-class boroughs run schools, collect taxes, and regulate zoning. (At least they can. Rural areas don't tend to concern themselves with zoning codes.)
Second-class boroughs are the most common. In addition to the above, they can maintain transportation systems, regulate animals, license day care facilities, and control pollution. (I'm not sure what this would entail at a borough level, but that's all state law says.)
First-class boroughs can do all of the above; the difference is that while second-class boroughs need to hold an election to acquire additional powers, first-class boroughs don't, in some cases. This is a relatively pointless distinction, and probably accounts for the fact that there are no first-class boroughs.
What determines what class a borough is? I have no idea, really; I can't find anything on it. It seems to just be a matter of how much responsibility the borough wants to assume.
Unlike counties, general law boroughs don't operate police forces. (Home rule boroughs do, however, and municipalities just have the city police.) Sheriffs are unheard of.
Any place not within a borough is, in theory, part of a single unorganized borough. In practical terms, there just is nothing there, and whatever duties a borough would do, the state legislature gets to do. (they don't like to do this, so the creation of boroughs tends to be encouraged.) If a division below state level is needed, the "census area" is used for these parts of the state. This is a statistical beast only, however.
Full list of boroughs (list complete as of October 24, 2002)
(Yes, all the names are geographic; we don't name boroughs after people. For those who wonder why, if there's an Aleutians East, there's no Aleutians West: it's a census area.)
Sources:
Title 29, Alaska Statutes http://touchngo.com/lglcntr/akstats/Statutes/Title29.htm (Title 29 deals with municipal government)
Alaska Community Information Database http://www.dced.state.ak.us/cbd/commdb/CF_COMDB.htm (for information on individual boroughs)