I am not an astronaut but the training they do is intensive. They spend years training to get to a point where they can go into space, they have to be very fit and healthy. Mission specialists will need to be experts in chemistry, physics biology, geology, or some other subject depending on the mission. They also have to be expert pilots if they are needed to fly the spacecraft.
It depends on their role on the mission! Flight specialists will have to undergo intensive flight training while mission specialists will have other roles. Sometimes astronauts have to do spacewalks on the International Space Station which is orbiting the Earth. This means that they will have to go OUTSIDE the cabin of the station and have absolutely nothing between their head and outer space. They are literally standing on the station and looking down at the Earth. This, of course is not a simple process and a lot of zero – gravity training goes into the preparation. This involves going into an aeroplane here on Earth and accelerating downwards at what we call “free fall” to simulate having no weight, just like in space. Astronauts have to be experts in various scientific fields (contrary to what the movie Armageddon will tell you 😉 )
During the time of the moon landings in the 1960s only fighter pilots and mad men could be astronauts. This has now changed as scientists are now needed for performing experiments on the International Space Station (ISS). So anyone can now apply to NASA or ESA (European Space Agency). However the process is still a hard one for getting selected. Like Joseph said, they select the best of the best. Each round of selections they receive 1,000s of applications and only a handful of people get selected.
If you want to be an astronaut it is probably the best time to be doing it as the next generation of astronauts will be headed for Mars and even to land on an asteroid!
To go to the International Space Station one typically requires at least 5 years of training as an astronaut. This is essentially because one needs to be a jack of all trades (scientist, doctor, technician). Further training will be required for missions to the moon and Mars.