Aikido is a modern budo (goshinjutsu: an art of self-defense) established by Ueshiba Morihei (1883-1969), and is one of the representative budo alongside of ancient Japanese martial arts jujutsu and kenjutsu.
Since he was a child, Morihei has trained in various bujutsu (martial arts) and in 1942 started to name and call his style of bujutsu, "aikido". After the WWII, aikido has spread more commonly across Japan when the art was handed down to Morihei's son Ueshiba Kisshomaru (although, it is said that Kisshomaru himself was rather reluctant about the disseminaion of aikido. Today, there are several schools of aikido.
The skills of aikido are suitable for facing both taijutsu (physical arts) and bukijutsu (weapon arts including swords and canes/sticks/staffs), and also assume situations confronting multiple people. In another word, aikido is an all-round martial art. Therefore, aikido was taken into the former army and navy schools as one of the subjects during WWII, and Morihei himself taught the courses. Even today, after the war had ended, the art continues to be taught in the Self Defense Force (SDF) and in riot police or the police special squad as skill-training, for it has great influence in manual martial arts and the skills when arresting a crime suspect.
As the founder's teachings, "Aikido is 70% knockdown blow and 30% throw" says, what makes aikido so unique is how you make use of the strength of your opponent to knock him down. When using aikido, you do not crash into your opponent but try to borrow his force and make it yours. It is basically grappling techniques but as you develop techniques you get to acquire throws and more offensive skills. Also, by using karadasabaki (body manipulation) you can make your skills more effective and efficient.
As compared to other budo, aikido puts great emphasis on the mental aspect, which the level will be reflected in the moves you make.This mental stage is called the aiki. This point is perhaps the most characteristic of aikido.
Fundamentally the art is centered on style exercises and do not have free form practices as in judo.
There are some schools such as "Yoshinkan" <http://www.yoshinkan.net/> which have English sites, and also some which you can have a trial practice. You can see a movie on the above site (written in Japanese but just click the "0" in the left-hand margin to view the movie) so those who are interested in aikido, have a look here.