battle mechanics?

Turn 3,cavalry (5) foward 52 killed archer 8,lost1
Archer (7) foward 22 killed cavalry 1,lost 5.

Turn 4,cavalry (3) wait killed archer 2 lost 0.
I lost.

My question is why did my archers move? And the archers didn't move backwards because in round 4 the cavalry didn't have to move.

You didn't command them by the looks of it...
When you enter a fight, goto the actions tab when the timer is on "00:00:00" then tap "command" then tap any unit you wish to command, they can move forward, backwards, stay still, attack different units etc.

your units are at the bottom of the screen.