Given that Deadpool as a superhero/anti-hero didn't really gain a lot of traction in popular culture until the last decade or so, he doesn't really have many recognisable or big-name arch nemeses the likes of Doctor Doom, Magneto or the Joker.
However that doesn't mean Deadpool is without his enemies, in fact as one of the most aggravating sometime-hero characters in Marvel comics he's had a long history of runs in with various villains and heroes alike.
We already know who our villain is going to be for the Deadpool movie, and the director Tim Miller has teased as to who might be the antagonist in the planned sequel.
Portrayed by Ed Skrein, Ajax will be the primary antagonist in Deadpool. Despite being perhaps not that prevalent a character in the graphic novels, he's important when it comes to Deadpool's origins.
"I think the rest of the world sees [Ajax] as a bad guy. Me, I think he's just misunderstood. His name is Ajax; or Francis. He went through the same Weapon X program as Wolverine and Deadpool." - Ed Skrein
Ajax, or Francis as he was formerly known, was an a cybernetically enhanced enforcer/guard at the Workshop - a lab which dealt with the failed results of the Weapon X Project. Wade Wilson was one of these failed results, kept as a prisoner to be experimented on by the mad scientist Dr. Emrys Killebrew. Killebrew performed the process that made Wade into Deadpool, which included having his nerves removed - ouch.
Wade and Francis always hated each other; Wade taunted Francis relentlessly (as we see in the Deadpool trailer) and Francis eventually tried to kill him by removing his heart. Of course, Wade just grew another one. Becoming insane in the process and taking the name Deadpool, he hunted down and apparently killed Francis.
However, like all good comic book characters Francis didn't stay dead, instead reemerging as the enhanced Ajax, with one hell of a vendetta against Deadpool.
Deadpool director Tim Miller has hinted that the sequel will contain Deadpool's enemy/sometime partner Cable (Nathan Summers).
"There aren’t really many definitive Deadpool villains, apart from Cable. If we don’t put Cable in Deadpool 2 I think we’ll be run out of town on a rail."
The son of Cyclops (Scott Summers) and Madelyne Pryor (the clone of Jean Grey), Cable is a character most commonly associated with X-Force and the X-Men but he's had run ins a plenty with Deadpool.
They first met when Deadpool was hired to attack the New Mutants, which included Cable as a leader/trainer of the group at the time. Deadpool has fought both Cable and the X-Force on various occasions, with varying outcomes between the two.
Often described as an anti-hero (like Deadpool) Cable was infected with a techno-organic virus which turns organic tissue into a mixture of tech and living tissue; he posses enhanced strength and durability as well as a techno-organic left eye and arm. He had telepathic and telekinetic abilities (though these have faded over time) and possesses the ability to time travel, though initially needs help from the X-Men to utilise this power.
Though initially enemies Deadpool and Cable did end up teaming up in later issues, for the short lived Cable and Deadpool run.