DOMS stands for "Delayed Onset Muscle Soreness" and it refers to the period 24-48 hours after a tougher than normal workout where it hurts to even stand up or brush your hair. I used to think this was a good thing, but latest studies show its a sign that waaaay too much muscle damage has been done and can actually be very serious or even fatal in extreme cases. That condition is called Rhabdomyolysis and its basically your kidneys shutting down because there is so much muscle tissue damage from the workout that the amount of waste products are too much. It's very serious stuff.
Yes, every workout involves some breakdown of muscle tissue. It's part of hypertrophy (enlargement of rebuilt damaged muscle fibers) and is a normal sign of progressively overloaded workouts. But in this case, too much of a good thing is not good.
BTW, it was thought that lactic acid was the reason for the soreness in DOMS however is turns out that its just the reaction of localized damaged muscle tissue. Lactic acid (waste products in the blood from sustained muscular contractions) might account for AOMS (acute onset muscle soreness) which is the "burn" you feel during exercise, but would be long gone by the time DOMS sets in.