Shadow Link is a mini-boss of the Water Temple in Ocarina of Time. Basically, he looks like you, except, umm... he's shadowy. Like many other Zelda villians, he has been recycled from a previous game, with various updates. Specifically, he was the final boss of The Legend of Zelda II (Link's Adventure), where he apparently was rather anticlimactic†, though, based on what I've read about him, he was quite similar in style to his 64-bit incarnation.
He appears in a room about 2/3 of the way through the level, which, when first entered, seems to be outside; this can be rather disconcerting, as the Water Temple is situated underneath Lake Hylia. The horizon is lost in mist, the ground is covered by a thin layer of water, and there is a large rotten tree right in the middle of the room. They even cut out the normal background music, making it seem really strange, especially since you have probably spent at least the last hour or two listening to it while playing through the first part of the temple. There are two doors, one in, one out (the out door being barred until you beat Shadow Link).
At some point your alter-ego will show up and try to kill you (he doesn't show up right away for some reason, you have to wander around a bit first). Unfortunately not only is he very fast, but he has some tricks that you don't have, and the only way to beat him (or at least, to beat him without getting seriously hurt) is to use some tricks of your own.
So, for killing him there are three major hand weapons: your Master Sword, the Biggoron's Sword, and the Megaton Hammer. The Master Sword is a bad idea for Shadow Link - if you swing it at him, he will either block, or dodge, or jump onto the end of your sword, standing there all cocky before smacking you on the head with his sword. This is really annoying, not to mention surprising. The hammer is OK, but it's fairly easy for him to dodge it, but while you're using it you can't use a shield, meaning you will be taking a lot of damage. So all in all I would say use the Biggoron's sword if you must engage him hand to hand, though it won't be a trivial fight. The longer reach of the Biggoron's sword will help out significantly, but his speed is nearly unmatchable (unless you're some 12 year old hyped up on sugar) so it still takes concentration. Each time Shadow Link is hit by anything, he drops into the ground to reappear in a different spot a few seconds later. Between this, his transparency, and the mist, it can be hard to keep track of where he is, adding to the fun.
Or (here there be hints) you can get all thinking outside the box on his ass and not fight him hand to hand at all. You may be saying now "Well if you're so damn smart why don't you tell us how, randombit?" Be happy to kids, it's a little thing I like to call the fire spell (technically referred to as Din's Fire). It takes a number of hits to finish him off, but in the meantime it keeps him away from you, so at the very least it's a good way to give yourself a breather. Basically you just cast it, wait for him to reappear, and then repeat until either he's dead, or you run out of magic. If you'd like to try your hand at beating him 'fairly', just keep this technique in the back of your head in case you screw up and just need to kill him before he kills you. I 'discovered' this method when I had two hearts and no fairies left, facing a nearly untouched Shadow Link - while dying in Zelda isn't that big a deal, it takes quite a while to get from the beginning of the temple (where you respawn) to where he is, so it's rather annoying to get killed by him.
You could also try using your bow, Bombchu, or the hook shot if you're feeling inventive, but I never had any luck using any of these items against him.
After you beat him, the mist and the tree dissolve to reveal a large room inside the temple, and the far door opens up, letting you continue.
† Props to amib for telling me about this, sadly I never got my hands on the NES Zeldas (Zeldii?)