As Jay said, the system is based around a pay-for-download model.
The Wii has wi-fi capability, and looks for a wireless connection (it can also be connected with a wire connection, like a computer). The Wii's main interface is a series of "channels" for the main screen. One channel is the one you select to play a Wii or GameCube game, another is the channel you choose to get the news off the internet, etc. (more info on this can be found on Wikipedia's page for the
Wii Menu).
The Shop Channel is where you look for games. It currently has a selection of almost 200 games for download, from the NES, Super NES, Nintendo 64, Sega Genesis, TurboGrafx, and Neo-Geo (games from other systems, such as the Sega Master System, are coming later this year). New games are added, in North America, every Monday. Games cost different amounts based on what system they're from, but they range from $5 to $10 (there's one game that costs, IIRC, $12, because it was a Japanese N64 game translated into English, "Sin and Punishment").
You purchase these with "Wii Points," and 1 point is worth $0.01. Wii Points are either bought directly as cards in a store (you enter the card's information when you're on the Shop Channel), or you simply buy them directly from the Shop Channel on a 1-for-1 basis - IIRC in blocks of 2000, (plus sales tax).
This model, in my opinion, is a great one. Each game comes with a digital reproduction of the instruction booklet, and is faithfully adapted from the original game (in some cases, there are unavoidable changes, such as some N64 games that used the rumble feature). You don't have to worry that there'll be problems with the game the way a used game might have, and you can effectively always save your game wherever you are by pressing the "home" button on the Wii Remote while playing. Between these downloadable games, and the fact that the Wii plays GameCube games, you essentially have access to every Nintendo console ever on your Wii, plus games from other consoles (such as the Sega Genesis and the Neo-Geo). I personally love the fact that I can potentially play Super Mario Bros. 1, 2, 3 and The Lost Levels (all for the NES; The Lost Levels was the Japanese-only sequel to the first SMB game, now translated and on the VC), Super Mario World (SNES), Super Mario 64 (N64), Super Mario Sunshine (GameCube), and Super Mario Galaxy (Wii) all on the Wii by itself.
To be fair, the system does have some minor drawbacks. First, while NES and some other games can be played simply by holding the Wii Remote horizontally, other games require that you use either a GameCube Controller (sold separately), or that you use the Wii Classic Controller (also sold separately) - personally, I recommend the Classic Controller for this, as it closely follows the layout of a Super NES controller, and is guaranteed to work with all Virtual Console games that can't use a horizontally-held Wii Remote (whereas a GameCube controller will only work with some downloaded games, and the button arrangement can be a bit frustrating).
There's also a limit to how many Virtual Console games you can have on the Wii at a time. The Wii has a grand total of 48 channels displayed (with 4 being used from the start, and somewhere around another six available for free download). Once your channels are filled, you need to basically delete Virtual Console games in order to load new ones. You don't lose games you've already downloaded - you can redownload games you've already bought for free at any time, but it can be something of a hassle. Also, you can only play games you've download on your own Wii; they can't be transfered to any other Wii system, even temporarily. Finally, the Wii's internal memory is roughly 500 megabytes in size; while most Virtual Console games barely make a dent in this, very large ones (such as most N64 games) can eat up a lot of size by themselves.
Bear in mind, also, that if you want to play GameCube games on the Wii, you can, but doing so will require both a GameCube controller, and a GameCube memory card. While the Wii does have internal memory, it only holds data for Wii and Virtual Console games. Also, the Game Boy Advance Adapter (which lets you play GBA games on a GameCube) does not work with the Wii.
For more information, check out Wikipedia's articles on the
Virtual Console and the
list of Virtual Console Games for North America.