Puffing up is similar to when a male bird puffs up to impress female birds, except when a human man "puffs up", it is to impress several other people with his air of importance. It usually implies a posture in which the person stands erect with his chest "puffed up".
Posturing is very similar, except the mental image can vary with the situation the man is in. Whatever physical positioning makes him seem the most powerful and self important is posturing.
Talking usually, though not always, accompanies a man who is puffing up or posturing, since this type of behavior is often associated with public speaking or speaking to an audience.