The Secret Base was introduced in the 2nd generation of Pokemon games (Gold/Silver). In those games, you pick a suitable place to have a sort of cave or treehouse 'hideout'. It had a 'Laptop' with the same functions as a PC, and there were several items you could use to decorate and customize the space. (My favourites were these floor tiles that play a musical note on a scale when you step on it.) Depending on where you chose for your base, it could be several different shapes. When you linked up with a friend, your secret base gets transferred into their game as well. If they find your secret base, they see all the decorations you did as well as battle you with whatever Pokemon you had at the time of the transfer.

I don't know all the details of D/P's bases, since I haven't done it myself (couldn't figure it out). I imagine it would have at least the same features. This time, instead of hunting around for special trees or mountain holes, you just go into the Underground and can make it there somehow. I mentioned Capture the Flag before--this is where it happens. You and another player try to find each other's secret base first and bring it back to your own. You can set traps and such all throughout the Underground to prevent their progress. Apparently when you win, you get some kind of upgrade for your base.

Lots o' stuff in this game.