JKR revealed this in a 2007 interview on the Today Show.
Harry and Ron work at the Auror Deparment at the Ministry of Magic, where Harry is the head of the department. Hermione is in some high ranking position in the Department for Magical Law Enforcement.
JKR: Harry and Ron utterly revolutionized the Auror Department in-- at the Ministry of Magic. So they-- I mean, they are now the consummate-- they are experts. It doesn't matter how old they are or what else they've done. So Harry and Ron lead the way in recreating the new Auror Department. And by the time-- 19 years later -- I would imagine that Harry is heading up that department, which is not corrupt in any way. It's-- it's a really good place to be. And Hermione ... I think she's now pretty high up in the Department for Magical Law Enforcement. Where I would imagine that her brainpower and-- and her knowledge of how the dark arts operate would really give her a-- you know, a sound grounding. So they're all at the ministry but this is a very new ministry. They made a new world.
A more recent article on Pottermore (transcribed version) clarified (and slightly changed) this further.
Harry is still an auror (or at least associated with the position enough to be referred to as one).
About to turn 34, there are a couple of threads of silver in the famous Auror’s black hair, but he continues to wear the distinctive round glasses that some might say are better suited to a style-deficient twelve-year-old.
Ron became an auror, but left the job two years later to become a co-manager at Weasley's Wizard Wheezes.
In the immediate aftermath of the battle Weasley, whose famous ginger hair appears to be thinning slightly, entered into employment with the Ministry of Magic alongside Potter, but left only two years later to co-manage the highly successful wizarding joke emporium Weasleys’ Wizard Wheezes.
Hermione is at the Department of Magical Law Enforcement, poised for promotion.
After a meteoric rise to Deputy Head of the Department of Magical Law Enforcement, she is now tipped to go even higher within the Ministry, and is also mother to son, Hugo, and daughter, Rose.