Wushu is the Chinese word for all the styles of Chinese Martial Arts. These arts can be Tai Chi, Shaolin Boxing, Wing Chun, and many other martial styles.
"Wushu" is the correct term for all Chinese martial arts therefore kung fu and wushu were originally the same. During the last thirty years, wushu in Mainland China was modernized so that there could be a universal standard for training and competing. In essence, much emphasis has been placed on speed, difficulty, and presentation. Consequently, wushu has become an athletic and aesthetic performance and competitive sport, while "kung fu" or traditional wushu remains the traditional fighting practice.
Karate was originally an Okinawan method of combat that almost completely dispenses with throws. Its blocks are hard and it is a power oriented style. Judo is a sport that involves primarily throwing and grappling. It is very similar to western wrestling and was invented in the late 1800’s by Jigoro Kano, in Japan. Tae Kwon Do is a Korean art, similar to karate, that emphasizes the feet as weapons and is also very power oriented.
Wushu, more commonly known as Kung Fu, is a Chinese self defense system that is comprised of both hard and soft styles. The Shaolin curriculum consists of many styles of Wushu that teach the use of throws, grappling holds, weapons, and self defense. It is therefore a more broad and complex system of combat than many other styles. Similar non-Chinese martial arts include Jiu Jitsu (Japanese) and Hapkido (Korean).
This is yet another result of western linguists confusing both eastern and western speakers. In the once near-universal Wade-Giles spelling, a Chinese "G" sound was written in English as "K", while what the Chinese pronounced as "K" was transcribed as "K' ". Thus if kung fu were supposed to be pronounced with a "k" sound, it would have been written as "k'ung fu." When Bruce Lee introduced American audiences to his martial arts, he both spoke and wrote the American "G", hence "gung fu." Confused? Don't worry about it.