Fenders keep you clean and dry when the road is muddy or wet.
There are 4 main kinds:
Painted steel, Stainless steel, Aluminum alloy (painted or polished), and thermoplastic.
Steel is least likely to crack, and less likely to dent than aluminum, but risks rusting. Painted steel fenders come with many cheaper bikes, such as beach cruisers, as well as on high-quality European city bikes, where the underlying metal may be stainless steel or galvanized to resist rust.
Stainless steel, polished fenders are beautiful and very durable, but expensive and slightly heavier than alloy or plastic fenders. They can also be more difficult to install.
Aluminum alloy fenders are common due to their light weight, and good looks. High quality polished alloy fenders look as good as stainless steel and weigh less. They are more easily dented or bent, however, and may crack after many thousands of miles of vibration. Some are difficult to install.
Plastic fenders are dent-resistent, flexible, and fairly cheap. They may be the best option on bikes that are often parked at crowded racks, and they are the most commonly available as an add-on. They are easy to install, because they easily bend into place and holes can be drilled with little effort.
In general, black (plastic or painted) and silver (plastic or polished) are the most readily available colors. Fortunately, silver or black will match almost any bike.
If you want fenders of a particular color to match your bike's frame, it is best to buy them with the bike. It is possible to have aluminum or steel fenders painted or powder coated to match the frame later, but it is an extra step that adds time and expense.
For full protection from spraying water, curved fenders should be at least 1/2 inch or 10 mm wider than your tires. 1.5" x 26" tires work well with 2" (50 mm) fenders, for example. If the tires are too wide, a little water may spray off the edge of the tire, but most will still be caught by the center of the fender. There should be at least 1/2 inch or 10 mm of clearance between the tire and fender, to prevent rocks, mud or snow from clogging things up.
Many plastic fenders, especially those designed for road bikes and "clip-on" finders, are not long enough to keep feet dry and the bike clean. The front fender should come and close to the ground as possible; those that have a rubber or leather mud-flap can come closest without limiting clearance. On the rear wheel, a short fender may not bother you, but it will spray mud and water on bike riders behind you!
Even worse are the flat fenders which have recently been stylish. Flat strips of bamboo or metal may look nice and "minimalist", but will drip off quite a bit of water to the side while you ride.
Full-length fenders, designed to fit your bike, will make it much more pleasant to ride when the road is wet and muddy, and will keep your bike in good condition.