If the universe keeps expanding, what is it expanding into? Will it turn to another big bang?

The Big Bang Theory basically states all the solid material in the universe was assimilated together at one point (a singularity), and then, under extreme conditions of pressure and temperature, a huge explosion resulted.  This explosion did what explosions generally do, ignite and expand outward, in a starburst-like expression. 

It is the general opinion of all astronomers to date our universe is indeed expanding, as if pieces fleeing in an outward progression from an internal blast. 

There are three theoretical predictions as to the outcome of this expanding universe.  In an open universe, with time being infinite, the outward expansion would continue, though slowed somewhat through gravitational force.  In a flat universe, the expansion would continue but would slow to zero after an infinite time while retaining a flat plane-like shape.  In a closed universe, the expansion would stop, then start contracting and recollapse back on itself again, possibly forming another singularity that could reenergize another big bang. 

The governing factor in the slowing in all three scenarios is gravity.  So yes, if the universe is closed, the reemergence of another big bang could be a possible future for the presently expanding universe.