When an airplane is in "straight and level" flight (wings level to the horizon/force of earth's gravity, and airplane neither climbing nor descending), all of the lift produced by the wings is directed upward (directly opposite to the force of earth's gravity, which is pulling the airplane down).
When an airplane "banks" (the wings are tilted in relation to the horizon - left wing up, right wing down = right turn, for example), a component of that lift is now causing the airplane to turn. That lift is "pulling" the airplane around the turn.
So what about the lift that was needed to offset the force of gravity? Now that some of that lift is being used to turn the airplane?
If nothing else is changed, the airplane will descend in the turn, exactly because there is no longer the same amount of lift to counter the force of gravity.
So, if the pilot wants the airplane to remain at the same altitude, he will need to add lift in the vertical direction (against gravity).
He does this by increasing the "angle of attack" of the wing. That is, he increases the pitch of the airplane (nose up, tail down) by pulling back on the control yoke/stick.
That deflects the elevator (on the tail of the aircraft) up, which pushes the tail of the aircraft down (nose up, also).
The wings are being asked to deflect more air downwards (and creating an even lower pressure area above the wings - that's how an airfoil works).
So, in essence, if the pilot wants to make a turn in level flight (same altitude), he banks the airplane, as well as pitches the nose of the aircraft "up" at the same time.
Turn the control wheel (left or right) and pull back on the control wheel a little bit at the same time.
I hope this is clear -
Monty