If you follow all of these steps carefully there's almost no risk to get hurt. And, if you'll get hurt when splitting logs, I'm not responsible for it.
UPDATE: Please read the comments in the bottom of the page, though these provide information on how to cut wood in a better and safer way.
Step 1: What you'll need
An AXE - Sharp, not too small, not too heavy.
A LOG - Dry, it's easier if it's quite small. If you got big logs read the part "Splitting big logs"
A STUB - Or something else to put under your log. It should be stable. The good thing with having a stub is that the axe doesn't get hurt if you fail to hit the log.
A SLEDGEHAMMER - May be good if the axe is stuck.
SAFETY GOGGLES - Good for the safety.
A FRIEND - Safe
If you got long logs you'll also need:
A SAW - To make shorter parts, like one foot (~30 cm) long.
A SAW BUCK - Or something else to hold the long logs with when you saw them