Karate originated on an island called Okinawa, which is a Japanese territory now. In the 1920's and 30's, it was taken to the main Japanese islands. There are a wide variety of "styles" of karate. Some emphasize long-range striking, some focus on close-range hooking and pulling techniques. Karate is, generally speaking, a striking art, meaning the main focus is on punches and kicks, and strikes with other weapons, such as the knee, elbow, wrist, or edge of the hand. Every karate style will have a series of "kata". Kata is both a singular and plural term; a kata is a collection of offensive and defensive sequences that look a bit like a dance. They are performed solo, although some schools will have two-person sequences. Sometimes, weapons are taught in karate, although these tend to be traditional Okinawan weapons and, if taught at all, are usually taught to those with a few years of experience.
A few examples of karate kata:
http://www.youtube.com/watch?v=jBSOGmIRoBs&feature=PlayList&p=029433BB6D5A6395&index=0&playnext=1
http://www.youtube.com/watch?v=LYKXrATefSQ
http://www.youtube.com/watch?v=DkV_2EWCSJY
http://www.youtube.com/watch?v=jcOyqNqeW98
Traditional weapons:
http://www.youtube.com/watch?v=Y8V--2H9lj0
http://www.youtube.com/watch?v=V8cpyWQs-YI
Hapkido is a Korean martial art that has its roots in Japanese Jujitsu, the unarmed fighting system of the Samurai. It is known for having a wide variety of locks and holds, especially wrist locks, as well as throws and sweeps. Hapkido incorporates striking into its system, although this tends to be secondary to its grappling. It has a wide variety of unconventional kicks. Sometimes, weapons like sticks and swords will be used in training, but like karate, Hapkido is primarily an empty-hand system. Unlike karate, solo forms like kata are generally not taught.
Hapkido:
http://www.youtube.com/watch?v=CVTAWLJCdOY
http://www.youtube.com/watch?v=MnNMCpOqyV0
http://www.youtube.com/watch?v=VMBQZ8FWcE8
As to which is better for self-defense, that's really up to how good the individual instructor is and how good the student is. A lot of people are looking for the "best" martial art, but really, there's no magic bullet. I'd suggest attending classes in both, and going with your gut instinct on what makes you feel the most comfortable.