A Capture Mission requires a player to find either Corpus Targets or Grineer Targets, incapacitate them, and capture said target before returning to extraction.
Upon the player's discovery of a target, the target itself will begin to flee the room and run towards a designated escape room. They will shoot back at Tenno with their weapons and occasionally drop Smoke Screens to vanish from sight, though the mission marker still appears on their body. On a rare occasion, they will also use a Rhino Stomp or in another name; Radial Blast attack to knock away players that get into melee range. If they are not caught promptly before they reach the escape room, the targets will escape and the mission will fail. Capturing a target grants 250 affinity to the player's Warframe. Up to two targets will be present, but only one target needs to be successfully captured to complete the mission.
Unlike Rescue missions, there is no need to escort the quarry back to extraction. Instead, upon capture the target will let out an agonizing scream while slowly disappearing, after which Lotus will state that the target has already been escorted safely to extraction.
If you use a stunning, high-damage ability (such as Mag'sCrush), you can kill the capture target, rendering the mission unwinnable. Be very careful with abilities if you are coming up to his room.
Be careful around Environmental Hazards when chasing the target, if he dies from a hazard (e.g. Fire Hazard) you may be unable to capture him, as the prompt will not appear. 
If two people press the capture button at the same time they might both capture the same target which would count as two captures. Doing this allows you to only have to actually capture a single target so you can skip the second target. If you do go and capture the second target the amount of targets remaining will drop to -1 and if two people capture the second one at the same time as well it will drop down to -2.
Sometimes, when capturing a target, you can spin around, creating a spiral of bright squares. This has no detrimental effect on gameplay.
As of Update 14, Capture Targets cannot have finishers performed on them, although they will still take bonus stealth damage if attacked under the right conditions (caught unaware from behind, or when forced to stumble after Ash's Teleport). There is currently no indication whether or not this is intentional.