How to activate animations when my character moves

When I put the third person controller script on my character, it moves but has no animations, even though I recorded animations for it.

How do I fix this?

You have to call animations with a script. For example:

if (player.isMoving() && !animation.IsPlaying("run")) {
    animation.Play("run");
} else if (player.isStanding() && !animation.IsPlaying("idle")) {
    animation.Play("idle");
}

You could also use the velocity variable of your character controller if this fits your requirements:

if (characterController.velocity != Vector3.zero && !animation.IsPlaying("run")) {
    animation.Play("run");
} else if (!animation.IsPlaying("idle")) {
    animation.Play("idle");
}

And another way would be using unity’s mecanim animation system.

Animation Scripting Reference: http://docs.unity3d.com/ScriptReference/Animation.html

Unity Mecanim Tutorial: https://www.youtube.com/watch?v=Xx21y9eJq1U