Trilobita is a class of extinct marine arthropods, that lived from the start of the 3rd stage of the Cambrian to the end of the Permian. The part of trilobites that generally gets preserved is the carapace, the heavily mineralized upper part of the animal.
Trilobites are divided into three sections, front to back -- the head, or cephalon, the thorax, consisting of separate and independently movable thoracic segments, and the tail, or pygidium, which often looks similar to the thorax but consists of a single fused piece. The thorax, and, often, the pygidium, are divided into three longitudinal lobes by two clear lengthwise indentations. It is this separation into three longitudinal lobes that gives trilobites their name.
Most trilobites had compound eyes, just as insects do now. (Some trilobites were blind, with no eyes.) Trilobites within the suborder Phacopina have what are called schizochroal eyes, in which a separate cornea covers each lens. In that case there are typically relatively small numbers of relatively large lenses, and they are easily seen. Other trilobites have holochroal eyes, in which a single cornea covers all the lenses. Usually the individual lenses cannot be seen in that case, but in some cases they can be seen under magnification. Trilobites in the order Agnostida are all eyeless and blind, and so are some specific clades within other orders, such as the superfamily Trinucleoidea.
In rare cases the softer parts of trilobites got preserved -- the legs and antennae. From these rare fossils we have a pretty good idea of what living trilobites looked like -- there were a pair of antennas sticking out from under the front of the head, and a pair of legs under each thoracic segment, and a couple of pairs of legs under the head. The legs were biramous -- for each walking leg there was an appendage above it with feathery fronds that served as a gill. This is a primitive condition for arthropods.
Because of their heavily mineralized carapaces trilobites were often preserved when other arthropods rotted away without a trace, so trilobites are over represented in the fossil record. Based on the number of fossils found one might conclude that trilobites were by far the most common medium to large sized arthropods up through the middle Devonian, but in fact they were probably outnumbered by crustaceans and other arthropods for most of that time. Trilobites are certainly the most common fossils on this site, but that reflects my collecting preferences as much as anything else.
For much more about trilobites, visit Sam Gon's trilobite site.