Hi, my name is Ben, and I play swarm.
Now, a lot of people will define swarming (in terms of wargaming) in what I consider the typical sense; that is, in terms of how you do it. You need a lot of cheap, expendable units, outnumbering your opponent by multiples, and heading accross the board in vast waves designed to drown his or her army by sheer weight of numbers. Typically, this is done by the usual horde army suspects, such as Skaven and Vampire Counts in Warhammer Fantasy, Tyranids and Orks in 40K, and Legion and Cryx in the Privateer Press Games. It's also not how I play. Side note: Why are most of these armies kind of gross and slimy? Weird.
I prefer to define swarming not by the method, but by the objective. A swarm overwhelms its opponent, providing him or her with too many confusing options. If your opponent attacks Units A and B, then he must ignore C, D, and E, which remain threatening. Likewise, if C and E are attacked, then A and B are still around and deadly. Of course, in a typical swarm of, for example, Skaven, there will be far more than five options. In overwhelming your opponent with choices, the idea is to promote bad decisionmaking. Like a man attacked by hundreds of bees, your opponent should swat around in a panic. Maybe you'll lose a bee or two to the swatting, but your opponent's confusion will let you eventually bear him or her to the ground. His guns might be bigger and better, but they're nothing if he can't make proper decisions on how to bring them to bear.
Vast, horde armies are a mainstay of modern wargaming, and they have the aforementioned strengths. They usually have typical weaknesses, such as lower leadership troops and lynchpin units or models that hold the rest of the army together, sometimes literally like with Vampire Counts.
The type of swarm that I prefer might not be as numerous, but it accomplishes the objective the same. In my current Dark Elf army, I like to run plenty of Dark Rider fast cavalry, some harpies, and perhaps a unit of shades or two. Using this setup, I can force an opponent to make multiple decisions in the first and second turns. He or she is generally unable to focus enough attention on the big hitters coming down the table, as they have to protect their weak bits against these fast units that are up in their face.
Like a typical horde, this army forces opponents to make decisions that have negative aspects no matter what, hopefully forcing a panicked "swatting" at my army. Weak spots like warmachines are taken out quickly, assassination runs can be taken at key enemy units, and hard hitting units like Knights can be pulled out of position. Disorganization and panic reigns while my heavy hitters (Cold One Knights, Black Guard, perhaps a Hydra) come in unmolested to finish the army off. I once had an opponent end up turning his two flanks, still on his side of the table, completely sideways to face one another in order to deal with all my little biting units. Realizing what he had done soon thereafter, he conceded the round.
All in all, we look as good generals to make an opponent fight the battle on our terms rather than theirs. Swarming them with confusing decisions, be either method, is a great tactic for doing so.