Not to sound snarky, but literally you just pour apple juice into a fermenter, add yeast, slap on an airlock, and wait. You can of course geek out and do a whole series of other steps/additions/techniques, but basically that's it. The result will be a bone dry, 6ish% cider. For sparkling, just prime with sugar or 3/4 of a can of applejuice concentrate and bottle/condition like you would beer.
If you want to up the game a little here are some simple ideas:
- Use the best quality juice you can get, either flash pasteurized or with ascorbic acid (vitamin c)
- Use Lalvin 71b wine yeast, tends to round out any malic acid from the juice, or Nottingham ale yeast, tends to play well with the fruity flavors
- 5 tsp of yeast nutrient in with your yeast, keeps them happy
- Stir or shake the carboy once or twice a day for the first week to 2 weeks. This blows off any sulfur so it won't effect the flavor later. It'll smell like rhino farts during primary
- Leave it alone. Cider takes longer to ferment and condition than beer, 2 - 3 months is not uncommon. This is even longer if you add sugar to bump up the ABV
Other than that, take a look at these forums, there's a lot of variations and techniques to try out. Have fun and let us know how it turns out!