A sperm cell has three parts: head, body, and tail. The body isn't much, just some protein that holds the head and tail together.

The head is like your standard cell body except that it contains no organelles, just loose DNA. It has a cell membrane, which is made of phospholipids with some proteins thrown in for structure etc, and a full layer of protein as protection. The most notable is the acrosome, a filament of the protein actin, which is also found in muscle. This is the part that binds to a receptor on the surface of the egg.

The tail is a flagellum, which is also a fancy array of proteins. It consists of microtubules in a "9 + 2" arrangement, which means there are nine pairs of them around the outside two in the center. Each pair if microtubules is connected to its partner by dynein, another protein, which pulls along the other microtubule like you would climb a ladder. Normally this sort of movement would make one tubule slide along the other, but these two are bound at the root, so they bend instead, making the whole tail bend. This is coordinated in such a way as to let them swim.