Abel is Street Fighter IV's equivalent of Makoto in Street Fighter III.
That is, a character that keeps you guessing on defence with his Change of Direction attacks and then when you're paralyzed into defending, he can move in with his Tornado Throw.
Abel sometimes struggles against characters who can keep him at distance, as his poking game isn't too strong, but when he closes distance he can end the match in seconds.
Change of Direction (quarter-circle towards + P, towards + P, towards + P)
Abel's Change of Direction punches can all be linked together, which makes for an easy but damaging combo when you cancel into them off another move (like crouching medium kick). You can also press towards + K to go low for a sweep, but there's not that much point to trying this. If the first two punches are blocked, don't go for the third. If blocked, it'll take a while before Abel can block again.
Wheel Kick (quarter-circle back + K)
Abel's overhead kick gets him fairly close but he bounces away from his opponent after he hits or is blocked, so it doesn't quite do the job. However, it's fairly safe on block and hits overhead, so it still has its uses.
Marseilles Roll (quarter-circle towards + K)
Very useful move. Not only does it get Abel out of the corner (particularly the EX version) but it can also be used to get past your opponent's pokes and projectiles, to get close and start using Abel's strong offensive options. You can be thrown out of this move though, so don't abuse it.
Falling Sky (towards, down, down/towards + P)
This air grab works better in combos than it does as a pure anti-air move, so don't rely on it when opponents show a fondness for jumping at you. The EX version can work but it's still a little inconsistent and a waste of valuable meter.
Tornado Throw (half-circle back + P)
Abel's command throw has decent range and you can use cr.LP to get your opponent to block before busting out the throw. After a throw, dash towards your opponent and jump towards him and press MK in the air. This will connect from the other side, which can be followed up with another cr.LP into command throw loop.
Heartless (quarter-circle towards, quarter-circle towards + P)
Abel's Super connects after pretty much everything he does and can be cancelled into off various moves - MP, his Change of Direction punches, cr.MK and so on. Abel has a flexible style of play, so you can choose to either save your meter for the Super or blow it on EX moves to do damage and annoy your opponent.
Soulless (quarter-circle towards, quarter-circle towards + PPP)
This is an extremely powerful move against projectile characters because Abel will run through the projectile and hit the opponent! This means even holding onto the Ultra will scare your opponent into not throwing any projectiles, which means you can approach your opponent with ease.
Basic Combo: cr.HP xx Falling Sky
Advanced Combo: Jump HK > cr.MK xx Change of Direction xx Heartless