Your veteran status is based on how many other games in the series you have played. Certain games offer rewards based on your Veteran status, Bad Company 2 for example gave you the M1 Garand for use in the multiplayer.
All Battlefield games onward from Battlefield 2 (expansions as well) count towards your status but I think there are only a few (maybe four like agent said) that actually make use of it (that being displaying your status in game or giving you a reward)
Originally I think you had to register the game to your account on www.battlefield3.com/veteran but now a lot of it is tracked automatically based on whether you have multiplayer stats on the EA servers.
Hope that helps :)