For real wood cabinets, your best option is often to restain and varnish. It's a difficult job, but is the most natural way to show off the wood. Oil varnishes and other clear-coats dry harder and with more luster than water-based clear coats. But oil-based products take make longer to dry---about 24 hours, as opposed to six or eight for water-based products. Oil-based products can sometimes yellow over time, so take these factors into account when deciding which type to use.
The refinishing process involves sanding the cabinets down to bare wood, then applying stain, letting it sit for a few minutes, and wiping off the excess. The clear top coat is applied in three or four thin layers after the stain is dry, with a light buffing with fine sandpaper between coats.
If the body or doors of your cabinets are made of laminate, you will not have the option of staining, and you may have trouble getting paint to properly adhere. You can buy new laminate that can be affixed to the cabinets using peel-and-press adhesive. You initially cut the laminate pieces slightly larger than the cabinet area, then cut off the excess with a razor. Make sure the previous layer of laminate is firmly affixed to the cabinets, otherwise, the new laminate will come loose.
Glazing is a refinishing term that refers to specialized stains and glosses that are used to create certain visual effects on your cabinets. For example, one type of glazing is an opaque gel that you spread onto the cabinets, let sit for a few minutes, then wipe off in a haphazard fashion so that some of the gel stays on the cabinets, collecting in the corners. This gives the cabinet an antique look. You can also get crackle paint, another type of glazing that is designed to split and crack once you apply it to the cabinets to give them an aged look. Glazing is as much an art as a process, so practice on some scrap wood first to get your technique right.
You may not usually think of painting as a form of refinishing, but it's a perfectly legitimate one. However, painting over fine wood like oak or cherry isn't usually recommended, so consider painting your cabinets if the wood doesn't have a natural beauty worth displaying.
Oil paints give cabinets a hard, classically lustrous look, but they are far more difficult to work with than water-based paints. Oil paints require longer drying times and paint thinner for cleanup, and you may find that the difference isn't notable enough to justify the extra effort. Whichever paint you choose, apply multiple thin coats rather than one or two thick ones, and always be on the lookout for drips.