Though Microsoft launched headfirst into online console gaming with Xbox Live and Sony fully jumping in this generation, Nintendo has been more reticent in their approach. Prior to Wii, just one console title utilized a modem: Phantasy Star Online I & II for GameCube.
The Nintendo DS has a rather strong lineup of WiFi-enabled titles, but like the Wii system there are friend codes that must be swapped by both parties in some fashion outside of the game. (Random online play is possible through a matchmaking system, but you cannot save a random opponent's information for a later challenge.)
A few more online Wii titles are coming on the horizon: Battalion Wars, Super Mario Strikers and potentially Metroid Prime and Smash Bros. Brawl. Notice anything missing? Various third-party developers have claimed that Nintendo early on had not provided code for online play, but we know Electronic Arts has managed to incorporate online into its upcoming Madden 08.
[Update: The details of Metroid Prime and Smash Bros.' online capabilities are currently unconfirmed. Wording modified to reflect that.]