Muscle cells are highly specialized cells that allow you to move. They are found in just about every type of animal. Not all muscle cells are the same. There are actually three major types of muscle in a person - each of which has a different structure. The function of all three types is approximately the same - they all allow movement. The structure and function of muscle cells is quite complicated, but this should provide a basic introduction to the topic.
I'll start with the type of muscle that you are likely most familiar with - skeletal muscle. These are the muscles that are on your arms, legs, torso, and just about every part that you can move. Skeletal muscle is controlled by voluntary movements. If you don't want to flex a skeletal muscle, it won't (there are some diseases which create exceptions to this.
The large structure of skeletal muscles are very different. They range from the very large muscles on your thigh, to the very small muscles in your hands - and everything in between. All skeletal muscles attach to bones by tendons. They move the bones when they contract. Skeletal muscle is so common and bulky that it accounts to about 35-40% of the entire mass of a person. That's a lot.
In addition to these three major categories, skeletal muscle is often sub-divided further. There is Type I skeletal muscle, which is also known as "slow twitch" or "red" muscle. These muscles are able to contract over a long period of time - but but not as strongly. They are used by distance runners to run for hours. Type II skeletal muscle, or "fast twitch", "white" muscle, is the other sub-type. These muscle fibers provide immediate power, but they tire quickly. They are the "weightlifter" muscles.
Smooth muscle is the second major category of muscle. These muscles are usually found surrounding major organs and blood vessels in your body. They are involuntary muscles - you cannot choose to contract or relax them. Instead, smooth muscle is controlled by your brain.
Smooth muscle lines organs like the esophagus (which actually has some skeletal muscle also), stomach, major arteries, bladder, intestines, uterus (only in women), and bronchi. There are several microscopic differences between smooth and skeletal muscle - they do not look alike under a microscope.
The last major type of muscle is cardiac muscle. As the name implies, this type of muscle is only found in the heart. Although cardiac muscle has much in common with skeletal muscle in terms of structure, it is involuntary. You don't have to think about contracting your heart. There are some specialized cells in the heart which spontaneously produce an electrical signal that spreads around the heart and forms a heartbeat. In other words, there is no signal from the brain for your heart to beat - it does it completely on its own (of course, it's a bit more complicated than that, but that's the basic idea).
Bear in mind that these descriptions of the structure and function of muscle cells is the most basic introduction. There is a lot to learn about muscle that cannot be included here. Entire chapters of textbooks are written that deal with these structures. Hopefully, this was a good summary and/or introduction however.