After completing a stage, there's a chance that you might've captured a Mercenary. Once in while you can capture two mercs at once (I got them once).
In normal Battlefield stages (including Secret Stages), most Mercenaries you fight are capturable: as long as the Mercenary has a 1* or 2* form in their evolution track, you can capture them. Currently, Mercenaries that are 3* or higher cannot be captured (outside of the Alternating Stages Soul evolution, see below). Old and new "evolution material" Mercenaries are not capturable.
Some Dungeons in the Special section of the Battlefield follow some different rules, but the Daily Battlefields follow the above rules.
Alternating Stages contain a special, more powerful (or valuable in the case of Bitter Root) Mercenary in each of their levels. Any of the Mercenaries which appear on the stage can be captured, per the above rules, but there is also a small chance that a Soul of the special Mercenary can be captured. Souls can only be evolved with more souls of the same type; when the Soul "level" reaches 10 (i.e. you've evolved 10 of the same Soul together), it evolves into the full Mercenary.
Event Dungeons, thus far, contain a specific Mercenary soul or two that can be captured from them, regardless of the Mercenaries that show up on their levels.
There is currently a discussion on the forums dedicated to finding more data about capture rates: