There's been a lot of threads on this subject, but in my opinion the term "session musician" is pretty wide-ranging. The common denominator is a musician who can play whatever they are asked to play by the producer/music director. You can be asked to play nothing but roots in 8th-notes with a pick on a P-bass with flats, or you can be asked to double on upright and electric and read a complex part while playing with an orchestra for a movie soundtrack.
Check out some of the guys who are the big-name "session" guys - Will Lee, Lee Sklar, Neil Stubenhaus, Nathan East, Michael Rhodes, Pino Palladino, Jimmy Johnson, Tony Levin, Anthony Jackson, James Jamerson, Chuck Rainey, Carol Kaye, Ray Brown, Milt Hinton, etc... Take a look at their discographies to get a glimpse of the versatility required to become one of "those guys."
Also, know that becoming a session musician isn't exactly the same as applying for any other job. Most often, it's a gig like any other - you are recommended to someone via word-of-mouth, that person likes your playing and keeps hiring you and starts recommending you to others, and it snowballs from their.