Lock main camera on x-axis

Your camera won't move unless you have somehow asked it to. Perhaps you've attached a script component, or it is parented to something which moves? These are significant choices. That's not to say they're bad, but you should be aware that you've made them.

I've found this is nearly inevitable for any project of appreciable size: you will probably need to write at least one script to manage your camera's position and rotation. For example, the script could assign the camera's position on every frame by matching the player's transform position on one axis only. The nature and scope of this script will depend quite a lot on what exactly you'd like the camera to do.

Manipulating transform.position, transform.localPosition, and so on will usually get you more or less what you want. transform.LookAt() can also be pretty useful. You can see all of this and more at the Camera script reference.

I realize this isn't a very specific answer. To solve your immediate problem, you could assign a particular value to the camera's position on the one axis every frame.

If you do write a camera control script, it's usually going to be one of the last things you'll do on any given frame; you can set a script execution order, or use LateUpdate().

I think this is funny; I gave my camera a rigidbody, parented it to the controlled game object, and locked the axes.