Karate was the first martial arts introduced in the United States and became well known. Thus people began calling all martial arts "karate", just like some people call all tissues Kleenex or all soft drinks Coke. So "karate" is a generic term that people use to refer to any type of striking martial arts such as taekwondo, kungfu, or karate.
Karate originated in Okinawa and Japan while Taekwondo was formed in Korea. Technically, karate uses more hand techniques such as reverse punch in their training versus taekwondo which uses more kicking techniques. Ratio of hand techniques to kicking techniques might be around 70/30, that is 70% hand techniques and 30 % kicking techniques. Taekwondo is usually 70% kicking techniques and 30% hand techniques. Taekwondo is known for its dynamic kicking techniques and has grown to become both martial art and Olympic Sport.
Olympic sport competiton has greatly improved the effectiveness of taekwondo techniques. Due to its techniques and organization, taekwondo has become the most popular martial art in the U.S. and in the world. Estimation is that 60 million people worldwide practice taekwondo. So if your cousin in Florida tells you that he is practicing "karate", chances are that he may actually be practicing taekwondo! Some grandparents call our taekwondo school the "best karate school in Lansing".