I've seen the movie twice and still didn't understand this. He was seemingly a member of Cobb's team who betrayed the team.
Who was the captured member of Cobb's team in Inception and why did Saito choose to capture him? Why didn't Cobb try to get him back?
His name was Nash, which is presumably his last name. He was the architect Cobb and Arthur had worked with before, and was working with them on their attempt to extract the information from Saito. Due to Nash's failure to create a believable dream level, they failed that mission. Cobol, the company that hired them, would be immensely angry with the team, having presumably paid them scads of money already. Nash was responsible for Cobb and Arthur now having to flee for their lives from Cobol, so I'm sure they weren't concerned about his survival.
As to why Saito captured him, the best answer is probably as a gesture of goodwill towards Cobb. Saito wanted Cobb to do the Inception mission for him, and as an attempt to earn his trust, offered the chance to get even with Nash for his failure.
Nash was the architect who designed the two dream environments we see at the start of the film; Saito's Castle/Dining Room and Saito's Apartment. It was his failure to correctly research the layout of the apartment (which was based on a real place, something that Cobb vehemently opposes) that resulted in Saito realising that he was still in a dream.
Fearing for his life after having been seen by Saito, Nash evidently approached Saito directly and offered to sell out his partners in return for being left alone. Saito, obeying the age-old maxim that you can't trust a traitor had his men beat Nash bloody and then offered to hand him over to Cobb to kill.
Cobb refused to shoot him, but neither did he offer to vouch for him (the guy's an incompetent and a traitor to boot) so he just stands by and watches as Saito's men drag him away. We learn from Saito that he's not going to be killed. He's probably just going to get another kicking before being thrown out into the street.
Nash, BEATEN BLOODY, sits on the far side, slumped against the window. Beside him: SAITO. He nods politely at Cobb.
SAITO: He sold you out. Thought to come to me and bargain for his life…
Saito’s BODYGUARD offers Cobb a GUN.
SAITO: So I offer you the satisfaction.
COBB: That’s not how I deal with things.
SAITO: Would you work with him again?
Cobb shakes his head. Saito’s BODYGUARDS PULL Nash from the chopper. Saito motions Cobb and Arthur to sit. The chopper RISES. Cobb watches Nash DRAGGED across the pad.
COBB: What will you do to him?
SAITO: Nothing. But I can’t speak for your friends from Cobol Engineering.
