Surfing Pikachu is hidden in Pokémon Stadium for the Nintendo 64. To teach your Pikachu Surf (a technique it can normally not learn), you need to first unlock the R-2 (Round 2) mode. Then do the following:

-Enter the R-2 Prime Cup's Master Ball division (you have to beat the first three divisions first).
-Assemble a team that contains a Pikachu that does not yet know Surf. You HAVE to use a Pikachu from your Game Boy game. You cannot unlock Surfing Pikachu using a Rental Pokémon. Do not register your team! Pick it directly from your Game Boy cartridge.
-You now have to beat the Prime Cup's Master Ball division using a three-Pokémon team that always contains Pikachu. Pikachu has to be in every one of the eight battles.
-You are not allowed to use any continues. If you do, you won't get Surfing Pikachu.

Once you have fulfilled all the above conditions, you are shown an award screen with Pikachu standing on a surfboard. From here on, Pikachu can use the Surf technique both in battle and during gameplay (Game Boy versions only). In Pokémon Yellow, you can now also access the hidden Surfing Pikachu mini-game. Simply go to the small hut on the northern beach of Route 19 and talk to the guy inside.