A center’s main roles are to carry the attack forward, and provide an impenetrable defense in the midfield. Both the inside #12 and outside #13 center should be attacking threats, able to break tackles and run through gaps while catching and passing at pace to set up the outside backs. It’s these two who provide the real go-forward for the team. If they’re able to get across the advantage line on the first phase, the team is immediately on the front-foot and the loose forwards will be able to exploit this advantage on the second and third phases..
As defenders, they need to prevent the opposition from doing the same thing. This means coming forward quickly (with the rest of defenders in one line) in the face of the attackers, making 1on1 tackles against equally big, fast ball carriers and then being able to re-align quickly for the next phase. Because most rugby teams defend with fewer players in the line than the attacking team, this requires the 12 and 13 to slide from covering one attacker to covering another. This is far from easy. In fact, it’s this responsibility for covering multiple attackers that make defending at 13 the toughest job on the field.
Excellent contact skills. Often the first tacklers in defense, and therefore are good at competing at the breakdown (competing for ball, slowing opposition ball down, etc.). Obviously good handing skills, and understanding of the game. Anticipation, vision of the field and ability to spot opportunities. They should be on the lookout for gaps, mismatches and overlaps.
The 2 centers 12 & 13 often specialize; one as the inside center closer to the flyhalf (#12) and the other outside center (#13), further from the flyhalf outside the inside center. The #12 sometimes plays a similar role to that of the fly-half (sharing the load). The #13 is generally the faster of the two and usually have more room to move. But quite often the 2 positions are interchangeable.