If a game is vanilla Diplomacy where all the players are identified, it is considered to be standard.
The most common variant is called gunboat, which simply means that you won't know who is playing which player. Because some people use the term gunboat to mean that players cannot communicate (this is a definition more common in snail-mail Diplomacy), some newbies get confused. Just think of Judge gunboat as anonymous and you'll be fine.
It is possible to combine gunboat with any of the other variants, so often you'll see games with two name variants. For example, you can play Fleet_Rome and know who the players are, or play Fleet_Rome as a gunboat game and not know who is the power behind the throne of each country.
There are generally two kinds of variants on the judge, "tweak" variants, in which a minor modification to the standard set up is made, and "board" variants, in which an entirely new board is used, often with some other minor rules modifications.
Common "tweak" variants supported by the judge are:
Fleet_Rome: Take a standard diplomacy set up. Remove the army in Rome. Place a fleet there in its place. Voila. Note the Fleet in Naples remains a fleet, giving Italy two fleets, like England. Debate is mixed about whether this helps Italy or not, but it does give the green guy a few more options.
Milan: Another attempt to help out poor Italy. The northern part of Italy is altered so that Piedmont touches Burgundy but not Tyrolia (and the name is changed to Savoy), Venice loses its supply center and becomes Venetia, and Milan (a supply center) is added between Savoy and Venetia. Rome and Tuscany merge into a larger Rome, which touches Savoy, Milan and Venetia. The former Venetian army is now in Milan.
Crowded: Take a standard diplomacy set up. Put a supply center in Ruhr. Put units on every dot (including the new one in Ruhr). Call the 4 dots in the Balkans a Power. Do the same for Scandinavia. Add Ruhr to the Low Countries and make that a Power. Combine Spain, Portugal, and Tunis into a 4th new Power and you have Crowded. Makes early diplomacy very important.
Common "board" variants include:
Hundred: Since Andy Schwarz designed this variant, he put it first, even though it is not supported by older judges (most of the current Judges do support it though). Hundred is based on the Hundred Years' War, and features just three powers. The game is swift and brutal, and works great for learning the judge syntax or playing when fewer than 7 are available.
For more info, check out the Hundred home page.
Loeb9: Kind of like crowded in that there are Spanish and Norwegian Powers, but there are several new neutral dots and provinces, some islands are passible, and there is an Arctic Ocean that must be vacant after each Fall turn.
Modern: A 10-player variant set in modern Europe, 1995, and one of the more popular judge variants, at least according to Vincent Mous, the game's inventor.
Empire: Another Mous creation, this one takes place in North America, and pits various regions in struggle for contintental dominance.
Classical Mediterranean As the name suggests, this 5-person variant deal with the Mediteerraneaum world in the Classical period, approximately 270 B.C. Carthage, Egypt, Macedon, Rome, and Syria battle it out in the epoch between Alexander the Great's death and the rise of Rome as Mediterrnean Hegemon.
Youngstown: Too hard to explain in words, Youngstown adds China, Japan, and India, as well as some colonies for the Europeans and uses a Eurasian map (with some Africa as well). The game has Off-Board Boxes to allow quick travel from some parts of the board to others.
Asia: This is a newer variant for 7 players focusing exclusively on Asia. Other than a new map and different powers, it is just like normal diplomacy.
Chromatic: Take 5 maps of Italy, mush them together around a central Switzerland and you have chromatic. It's a fully symmetric 5 player game.
Machiavelli: Really this is a totally different game, with paid troops, famines, plague, etc. It was causing some problems, but the new judge code supports this very fun game again. The rules are 20 pages long, so you'll have to get them from a judge. Try: get rules.machiavelli