I was wondering how MMO battles work. I do not play any MMOs so I don't know. Is it just a bunch of people running around fighting? Or do players group up and use tactics and strategy?

Also is the combat going to be similar to past TES games or will it be the typical lock-on with auto attack with skills?

In an MMO there's as much variety in the combat as players are playing the game. You will be more successful if you get into a group and use tactics and strategy to play. That's why I always recommend finding a good guild that suits your needs. But you will still be able to participate and be effective in one on one combat, small groups, or "zerging" as it's called when a mass amount of people just rush an area. The combat will be similar to that of previous MMO's, with a twist and what i believe will be a refinement of the traditional hotkey combat.

I recommend you read the article http://tamrielfoundry.com/development-faq/ for the most up to date information about the game.