Rotate around object's own axis

In Unity when I change rotation values in ‘Rotation’ fields, for X axis it rotates around object’s own axis just as I want but for Y and Z axises object rotate around something else’s axis. How to fix it?

I’m not entirely sure what you’re asking but you could just parent it under the object you’d like it to be relative to.
If you want to do this by code you could use something along the lines of this:

public class RotateAround : MonoBehaviour
{
    public Transform parent;
    public float angle;
    void Update()
    {
        transform.RotateAround(parent.position, Vector3.up, angle * Time.deltaTime);
    }
}

I’m assuming that you got a script similar to this :
transform.Rotate(1, 1, 1);
the first number is the rotation on the “x” axis,the second on the “y” axis,and the last on the “z” axis.
You could simply set the rotation values to 0 something like this :
transform.Rotate(1 (or whatever number you want) , 0, 0);