When you split, each hand is treated as a separate hand, and played like any regular hand. The outcome of one hand does not affect the other. Most casinos allow you to split up to four times (like if you split 8s and get another 8, split again.)
If you've got good rules going, you can double your hand after your split.
Usually if you split aces, you only get one new card on each hand. It's not a good rule if you're not allowed to re-split aces.
Hope this helps. Happy playing.