Many game's rules come with variants explicitly described within the rules themselves.
Some rules describe the variants beside the relevant parts of the rules, while others list all suggested variants at the end of the rulebook. I favor the latter approach, as this make the basic rules easier to read.
Types of Variants
Additional rules for different number of players: Provided to allow the game to be played by more or less players than can be handled by the "standard" setup. These are always welcome.
Additional rules for younger players: To allow younger players to enjoy a more basic, easier, or quicker version of the game. These are always welcome.
Longer or slower play: To provide an explicit mechanism for playing a longer or shorter version of the game. These are welcome, but I've rarely seen both of them used in the same game group. Unless used as an introductory game, I've never seen an explicit variant for a shorter game that didn't cripple the game. As to longer variants, these are used if they correct an undesirable element of the original game. For example, auctions in Tikal take slightly more time, but correct the luck of the draw in the original game.
Tighter or easier play: For instance, in Evo you can open one less gene than the number of players, which means that one player will end up without a new gene at the end of the bidding. This makes for a "tighter" game, i.e. where you have more player contention and must manage your money more carefully. These are welcome, and will be enjoyed more or less depending on the game group and the type of game experience they are looking for.
Alternate victory conditions: In some cases, an alternate victory condition makes for an easier or tighter game. For instance, in Peg Solitaire, you can win with one stone remaining, or one stone remaining in the center hole. The latter is a tighter (more difficult) variant.
On the other hand, in my game It's Alive, changing the victory condition changes the strategy of the game: from buying all the cheap cards in the basic version of the game to auctioning the more expensive cards in the advanced version.
Imagine, for example, that the victory in Soccer were decided based on who had the least goals rather than the most goals. The strategy of the game is significantly changed (actually, the game doesn't work at all, which is why the goal of the game is so important to the play.)
Variants for alternate win conditions must therefore be thought through very well. The odds are very strong that most people or groups will like one and not the others.
Equal but different play: The Last Step Game, which I just reviewed, offers dozens of alternate rules for piece movement, in order to spice up the game and to avoid you having to play the exact same game over and over. Assuming that all variants have been tested, multiple variants are welcome, but, again, most people will like one or two a lot more than the others.
With different play variants, you run into the situation of: only one variant at a time, or mix and match? With N variants, you can have N^2 different games. And if the variant simply says "try scoring more or less points for the foobar", you are thrust into the role of game designer, hopefully in a safe framework where whatever you choose will still make for an interesting game.
Extra pieces: In most games, I don't use the extra pieces. For example, I have yet to play with the extra cards in Lord of the Rings: the Confrontation. Maybe it's just as good a game, maybe it's a better game, but I'm reluctant to try it.
On the other hand, it may be that the extra pieces make a better game, in which case I'll never play without them. In which case, why weren't they part of the original game, with a variant of not playing with them?
One issue that arises with extras like extra cards, is that the basic deck of cards eventually shows wear from frequent play, while the variant cards maintain a fresh look, making them difficult to fold into the basic deck without them standing out.
In my opinion, keep extra and optional pieces to a minimum; better variants make use of the pieces that are already included in the basic game.
Let's look at the bad and good of explicitly described game variants.
Variants can feel like laziness on the part of the game designer. "What, you couldn't play-test just a bit longer to find the best game?" Instead, the designer makes the players do the final work. In fact, I believe that this is sometimes the case.
If the game includes multiple variants, the odds are that many of these will be less well-received than other. The designer risks that his players will play the ones they don't like first, and never find the one they would love.
If variants are significantly different from the base game, they may be better received if released as an optional expansion purchased separately. The variant may add unnecessary pieces to the game. They complicate and lengthen the rules.
Finally, when players describe, rate, and rank a game, what game are they describing?
Explicit variants may provide a means for more or less players to play a game. For instance, if the game is created for 3 to 5 players, a few changes to the rules might make the game enjoyable for 2 or 6 players. Rather than force players to figure out how to do this, an official variant assures the players that the designer has vetted the proper balance for the game for this number of players.
A variant can provide a simpler game for younger players to play, if the game is otherwise complicated and generally suited for adults or older children. Different variants can appeal to different types of gamers: younger, older, or serious versus casual play.
Variants provide multiple ways to play a game, keeping it fresh. If many variants can be combined in multiple ways, you may never play the same game twice.
The inverse of the problems listed above is that players may find one variant more enjoyable than others. Rather than the designer forcing one explicit way to play a game, a variant lets the players choose their preferred version, making more people happy.
And explicit variants may inspire players to create additional variants the designer didn't think of. This may have the effect of turning the game into a game system.