Conventional bike tyres and mountain bike tyres are "clincher" type, also known as "wire-on." They consist of an outer tyre with a u-shaped cross section, and a separate inner tube. The edges of the tyre hook over the edges of the rim, and air pressure holds everything in place.
The "bead" is the edge of the tyre. On most bicycle tyres, the beads consist of hoops of strong steel wire. The beads are what holds the tyre onto the rim, and are, in a sense, the "backbones" of a cycling tyre.
Cloth fabric is woven between the two beads to form the body or "carcass" of the bike tyre. This is the heart of the tyre, the part that determines its shape. The vast majority of tyres use nylon cord, though some use other polyamides. The fabric threads don't interweave with crossing threads as with normal cloth, but are arranged in layers or "plies" of parallel threads. Each layer runs perpendicular to the next layer.
Once the fabric has been woven between the beads, and the tyre has its basic shape, it is coated with rubber. The rubber is mainly there to protect the fabric from damage, and has no structural importance.
The rubber that comes into contact with the ground is called the "tread." This area usually has thicker rubber than the "sidewalls" of the tyre, mainly for wear resistance. Most cycle tyres have some sort of 3-dimensional pattern moulded into the tread, which may or may not enhance traction.
Many bicycle tyres are "gumwalls" or "skinwalls." Gumwall bike tyres have tan sidewalls. This makes the sidewalls slightly more flexible, reducing rolling resistance. Skinwalls have either no rubber on the sidewalls, or a very thin layer. This, too is an attempt to make the sidewall more flexible and reduce rolling resistance.