Why did Concorde not have horizontal stabilizers?
(Source: concordesst.com)

It does have elevators in the form of elevons at the trailing edge.  A delta is effectively a tailless flying wing with a really long chord.  Like any flying wing, pitch stability is achieved by down force generated at the local trailing edge by a control surface that does the same job as a regular elevator/stab, by applying down force to balance the forward pitching moment of the CG ahead of the centre of pressure (an airplane is really just a teeter totter with the C of G at one end and the elevator at the other).

Fuel transfer is used to keep the C of G in a sweet spot range to minimize trim drag, which is the penalty taken in generating the down force at the rear, so that the down force can be no more than necessary for adequate pitch stability.

As for why they didn't use a horizontal stabilizer, the reason is that the delta plan form with elevons provides the required stability and control in pitch and allowed the designers to dispense with a separate tail surface, greatly reducing the drag of a separate tail (not eliminating, because there is still a drag penalty from the elevons generating down force).  Same reason anybody goes with a flying wing.  Plus, you need lots of sweep for low supersonic drag and lots of wing area for high altitude efficiency, so a tailless delta fits the bill.  

A limitation of this is a narrower CG range (a more limited forward CG than with a regular tail that is farther aft and has more leverage) because the elevons have a short moment arm, a limitation of all flying wings.  The ability to transfer fuel to adjust CG would help with this and I think Concorde's modest passenger loading didn't cause the CG range issues that a normal airliner would have to deal with.