The easiest way to begin to understand the game of MMA is to examine the basic anatomy of a fight. Like boxing, both competitors start standing up and make the most of the striking arts; kicks of all different kinds, punches to the head and body, and when slightly more inside, elbow strikes to the head and knee strikes to the head and body, including the occasional crazy "flying knee" a la Sagat from Street Fighter.
Now, most people don't find getting punched, kicked, kneed or elbowed a whole lot of fun, so after a while a fighter will usually opt to take the fight into the "clinch," which is considered the second basic stage of a fight. Where boxing referees will separate the fighters when they get too close together. MMA fighters will fight from the clinch, generally by using it as an opportunity to control the opponent's body to unleash even more debilitating strikes from in close, or opt to slam, throw or otherwise bring their opponent to the third combat phase - the ground.
Ground fighting is where MMA most greatly deviates from boxing and most other martial art competitions, and it is where the game of MMA is at its most elegant and most vicious. Early UFC events saw plenty of wrestlers fight plenty of stand-up fighters, and the results were almost always the same. While the striking specialists could usually get off a few good shots, they were never well trained enough in any grappling to be able to defend against a good wrestler's takedown efforts, and would inevitably end up on their back getting punched in the classic schoolyard-bully-beating position, the "ground and pound."
Brazilian jiu jitsu fighters, however, train to win fights from their back by using the leverage of their body to sweep their opponent from the top to the bottom and taking the dominant position for themselves, or to catch them in a submission hold from their back and leave them unable to continue the fight for fear of breaking something or being choked. While the ground-fighting often leaves the layman bored, those who have practiced any submission wrestling of their own know to look for the little things - a slightly exposed arm or the briefest hint of poor posture - that can reverse a fight's momentum. No matter how much has happened in a fight, anyone is liable to be knocked out or caught by a submission that would change the tide of the match.