Tailplane/Equilibrium conditions
To be statically stable in pitch, the tail has to generate a downforce that opposes a continuous nose down pitching moment.  Everything else being the same, more speed increases downforce, less speed decreases it.  The equilibrium state is when the opposing forces are in balance in steady state flight, hands off the controls, and the airplane is said to be "in trim".  This is achieved at a given airspeed for a given set of "configuration of forces" you might say (pitching moments, thrust lines, C of G positions etc) and is called "trim speed".

Trim speed is the speed at which tail downforce balances the nose down pitching moments to achieve equilibrium (not accelerating or decelerating) for a given set of physical conditions.  FAR 25 static stability requirements require the plane to positively seek this speed by pitching uphill or downhill when disturbed from that speed in an attempt to regain equilibrium.

So the relation to the tailplane is that speed in excess of trim speed causes the nose to pitch up due to excessive downforce, which makes it go uphill and slow down, trying to get back to that equilibrium state.  Vice versa when speed is less than trim speed; it pitches down trying to get to the equilibrium state.  In an actual airplane it overshoots its equilibrium point several times trying to get there and eventually zeros in (the Phugoid oscillation).

Some other configuration change, a power change, a C of G change, a dragging item change, a flap sticking down, or elevator incidence change, will alter the speed at which equilibrium is achieved or trim speed. Once stabilized at a new trim speed, it will now seek the revised trim speed as before, if disturbed, as the tail responds to downforce changes with airspeed as described.