What is a bear hug? Let's start with what it isn't: gentle.
Bear Hugging for Dummies
A bear hug looks like a normal hug, but laced with enough testosterone to kill an elephant. This is a move used to injure or dominate an opponent, especially in a grappling match. As opposed to a hug around the neck, this move is executed by wrapping your arms around the other person somewhere between the waist and armpits. This can be done face to face, or Brokeback (ha!) Mountain style. Then you squeeze as hard as you possibly can. Brute force alone can easily crack ribs; pressing your forehead into their sternum can cause intense pressure that will hopefully stop any of their aggression allowing you to press home the attack or retreat. If you're particularly brawny bastard, you can break their back and incapacitate them below the site. It is not necessary to restrain their arms in this maneuver, but it will help you to not get knocked the hell out.
If you are taller than your opponent, or strong enough to hoist them off the ground, you can exert a lot of control over them. If you can keep their feet off the ground, well, they can't go anywhere. If you also have their arms pinned, they have very few options until you put them on the ground. I'll leave the forcefulness of that up to you. Fighters use this technique to slam opponents to the ground as a precursor to a submission hold, or chuck their opponent out of bounds for a quick victory.