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Basics of Taming
You cannot tame other Tamer's monsters as you fight them. Also, you cannot tame an angered mother monster upon entering a battle with one when trying to pick up a wild egg.
Upon taming a monster, it will be added to your party, provided you have a slot available. If not, it will be added to the next available slot in your monster storage boxes.
Whether a monster will accept the pact you offer it is dependant on several variables. You are more likely to tame a monster if the monster's in your party have levels around, or, ideally, higher than the monster you are fighting. Weakening a monster so that it has low HP also makes it more likely to accept your pact.