In order for a veto to be overturned, two-thirds of the House and Senate must vote to overturn it. A bill is returned to the House, where it originated once it has been vetoed with information on why the president vetoed it. If the House does not get the two-thirds needed to overturn the veto, it will not go on to the Senate.
The president has ten days from the time he is given the bill to decide if he is going to veto it or not and send it back. The Congress has the remaining time left in that session to schedule the vote and see about overturning the bill.