Converting a unit changes its control from one player to the other. If you convert a unit to fight for you, this can also move your frontline forward. This has no other immediate effects, and the unit will stay in the same place it was before.
Units keep everything the same when they are converted: strength, abilities, status effects, etc.
Visually, convert is represented with a marionette control bar and pink particles. The converted unit changes directions and colors to signify its new ownership.
Convert is a very strong effect that can create massive board swings. But for some cards, it can also be a downside.
Steal strong enemy units or enemy units with strong abilities.
Convert an enemy unit far away to gain massive frontline.
Counter cards with on-death abilities and prevent them for triggering for your enemy.