There are several different materials used in countertops, and each is treated differently when it comes to repairs. The most common is plastic laminate, usually referred to by the brand name Formica. The second most popular countertop material is generically known as solid surface, but often referred to by the DuPont brand name Corian.
The abuses these surfaces are subjected to are numerous and cause different types of damage, which are dealt with in different ways. I'll touch upon the most common.
Laminate Problems and Their Fixes
Many of us have plastic laminate countertops. They're durable and inexpensive. One of the most common problems with them, though, is the failure of the glue that binds them to the substrate. This typically happens at the edges and seams. If the bond is weak at any point, moisture can invade and start the process of delamination. If caught early enough, before the damage spreads or before contaminants, such as dirt or food, infiltrate, they can easily be repaired with the proper type of adhesive. Laminates are initially bonded with contact cement, which is what you want to use if a fairly large area is involved. However, many experts recommend GOOP brand glue for smaller areas such as edging. Clean the area, apply the glue per instructions, and tape the piece in place while the glue sets. If it's a flat horizontal area, weigh it down with phone books. Cracks and tears can also be reglued. Always read the instructions with these types of glues, as most function differently from each other.
Gouges and scratches can be repaired with colored epoxy or other similar products. Slight surface burns can sometimes be scrubbed away with a mild abrasive cleanser. Serious burns usually require replacing the surface altogether. Sometimes you can resurface laminate over laminate; sometimes you can remove the old laminate and replace it; and sometimes you just have to replace the entire countertop, which isn't as painful as it sounds. Often you can buy premade countertops at your local home improvement store, which can be less expensive than a repair.
Solid surface countertops are completely different: they're far more durable, which means they last much longer. They cost more, but there's savings in the long run. They don't burn, light scratches can be sanded with a fine grit wet sandpaper, and gouges and chips can be filled with paste-type products, much like filling a fender dent on your car. Breaks can be easily reglued if they're small enough or unobtrusive. However, a large break might require the touch of a professional in order for the repair to be invisible.
This is a broad subject with a lot of "sometimes" involving a lot of different solutions, many of which are easily tackled for little expense. As with most home repairs, common sense is key, and when in doubt, ask an expert.