Burke and Stryker are agents with the CIA.
More specifically, they are the two members of a small department known as Red Cell tasked with "thinking outside the box" in their analysis, "to find the possibilities that other analysts might overlook or dismiss". If a solution seems impossible to come up with or if the facts do not meld together to make sense, the personnel from Red Cell are called in to give it a look.
Burke is Jonathan Burke, a seasoned veteran of the Agency who spent his time in the sandbox and has the war wounds, physical and mental, to prove it. As the senior member of the team, he had once had a fair number of people under his command but his persistent dourness and biting comments, coupled with his lack of empathy when a mistake would happen, drove all beneath him to seek transfers. When we first meet him, he was the Red Cell. As an analyst, his skills are exemplary; his people skills not so much.
Stryker is Kyra Stryker, a relatively new member of the Agency but already battle-tested in her own way. Shortly after graduating from the Farm, Stryker was sent as a field operative to Venezuela and had been there a year learning the ropes and doing quite well at it. Then the Chief of Station, a political appointee rather than an experienced agent, made a monumentally bad decision and Stryker was shot in the line of duty and nearly emprisoned for life. Now back in the States and needing someplace to work, fearing a firing was in the works, she is shunted to Red Cell. Was it punishment or a last-minute reprieve; she was not sure.
You would figure that mixing someone with such lousy ability to interact with others, like Burke, and someone with a earned chip on her shoulder and a desire to not be behind a desk would be a bad idea. Kathryn Cooke, new head of the CIA, though, had other opinions and it was she who forced the pairing. And she was right.