Can the Nintendo DSi play Nintendo DS games?

Yes, the Nintendo DSi can play Nintendo DS games. The NDS and NDSi are compatible with each other and can link for multiplayer as well. There are really no such thing as 'DSi games' except for DSi-ware and one or two games that require the DSi camera, none of which are compatible with a DS.

However, since the DSi has no GBA slot, Game Boy Advance games and DS games that require this slot (e.g. Guitar Hero: On Tour) cannot be played on a DSi. Flash carts and unofficial hardware won't work either, but there may be a newer version available, designed for the DSi.

Another problem you might find is that if you have any Wi-fi capable games, they will only work with a WEP security enabled network (or of course a network without security), not WPA.