Every professional sports league needs a way to fairly and equally disperse new, young talent into the league to promote parity. Almost all pro sports leagues use an annual process called the Amateur Draft to accomplish this.
An amateur draft is a draft where technically anyone and everyone who is not under contract of some sort with a professional team is eligible. Most of the time, this consists of young players between the ages of 17 and 22. In most instances where "the draft" is being referred to in professional sports, the amateur draft is the one assumed.
Each league has its own rules and variations, but almost all order by reverse-order-of-finish in the previous season in some method or another (for example the NBA adds an extra wrinkle to this with its Draft Lottery among the non-playoff teams from the previous season. Thus, a team with the worst record the previous season isn't necessarily guaranteed the first overall pick).
The MLB Draft has gone the extra step of technically being known as the First-Year Player Draft in order to close some loopholes in the previous system that allowed some draftees to "dodge the draft" by playing for a rival or independent league instead of the team that drafted them.
The NFL draft focuses almost solely on college players, while the NBA and Major League Baseball consider both domestic college and high-school players -- (overseas players aren't drafted but signed as free agents in MLB, but can be drafted in the NBA). The NHL draft covers a broad spectrum of high-school, junior hockey, college, and overseas players in their draft pool.
The annual NFL Draft is probably the most-heralded of the 4 major sports drafts, while MLB's is the longest and most-complex while recieving minimal media coverage. The NHL Draft recieves good coverage in Canada and regional coverage at best in the United States.
(Special acknowledgements to Orange Julius for clarifications and notes regarding the MLB Draft.)