Chekov is played in the new series of Star Trek movies by Anton Yelchin. He was born on Mar. 11, 1989. The character was introduced in the second season of the original TV show to add a bit of youth to the cast.
Having a teenager on a military ship was actually one of the more realistic aspects of a show that was filled with weird aliens and laser guns. The US military will accept recruits as young as 17 years old, as long as they have their parents' consent.