An 'accessor' method is simply a 'get' method - a method you call to access some value(s) in an object.
A 'mutator' on the other hand is simply the opposite of an 'accessor' method, i.e. a 'set' method, it allows you to mutate or change the contents of an object.
If you allowed x to be public then any class would be able to read or alter the value of x, and you would be exposing the implementation of the object. For example it may be that x is a temperature that is held as a float in the class. The class may implement the temperature within itself as the number of degrees above absolute zero. Now what would happen if another class set this to a negative value ? How could the object prevent itself from having nonsense values placed inside it ? What would happen if we decided to re-write this class so that the temperature was held as degrees centigrade - we would have to go looking through all our other classes to find any that set the temperature x. We lose all the advantage of OO and give ourselves the headache of traditional procedural programming.
Now by making x private we hide it from all other classes and we provide methods to set and get the temperature e.g. public float getXFarenheit(); or public boolean setX(float centigrade);. This is known as encapsulation. So, as Christopher says, within these methods we can do any amount of checking and conversion we like and hide the details of how x is actually implemented in our class. Also, since every other class has to use these methods to get at the temperature we only have to change one class if we change the implementation of x (providing we still use float as the parameter and return type).
If you look at the API
doc in the documents part of the JDK you will see very few of the Sun supplied objects (class Point is one) let you see how they hold data internally, almost everything is accessed through method calls, e.g. class Vector has a method size() to return an int representing the number of objects contained in the Vector. This is good practice.
Hope this helps...