Your brain will take over after a while and bring you out of the initial stupor you plunge into. This can take anything from a few hours to a few days.
If you were the one who was dumped, it's normal to find yourself asking if it's really happened and struggling to accept the inevitable - it's always tough.
You'll find yourself thinking: "He's made a mistake, of course he'll come back to you, there's been a misunderstanding, all's not lost," etc etc. It might be a comfort to you at first, but what you're doing is denying the reality of the situation. Even if reality doesn't exactly seem like your best friend right now, you have to accept it. If you keep harboring delusions of getting back together, you'll only make it harder on yourself and slow down the healing process.