Does an axis of symmetry determine chiralty?
Chirality is left hand—right hand. The left hand is a mirror image of your right hand. If you take this mirror image and then rotate or shift it and you are able to superimpose it onto the original image, it is achiral. This is not possible with the hands, the thumb will always be wrong.

A plane of symmetry is sufficient but not mandatory to give an achiral compound. But it should be clear that mirroring an image that contains an intrinsic mirroring plane will give the same.

Another sufficient but not mandatory symmetry operation to generate a chiral compound is inversion. If some compound is point symmetric, it is never chiral.

And there are other elements of symmetry that are rare if not accompanied by planes or centres of symmetry but which are sufficient for generating achiral compounds: Improper rotation. In a nutshell, this is a symmetry element that requires a rotation with a subsequent mirroring perpendicular to the rotation axis. Take a tetrahedral compound such as methane: If you select the correct axis, rotate by $90^\circ$ and then mirror it perpendicular to that axis, you will receive the same product.

Strictly speaking, planes and centres of symmetry are special cases of improper rotation with a rotation of $360^\circ$ and $180^\circ$, respectively. So one could say that improper rotation is sufficient and mandatory to give an achiral compound.