Clarity On Overriding Equals Method

For a few days I've been reading about the importance of overriding the equals method. How overriding it actually determines or checks the values stored in the variable. I realize that you can check the values stored in the primitive datatypes with "==", and when you don't override the equals method it acts the same way, right? When used with a reference datatype, "==" or the default equals() method only compares, or sees, if the variable is pointing to the same instance of a class. For some reason, in the examples, what is taking place to actually check the values stored inside the variables.

Here is part of an example (I've added comments for things that are confusing me):

@Override
public boolean equals(Object obj) {
//So we use Object here instead of the class type
// we're overriding this equals method for?
Is this so that we can use it to check different types? (overloading?)
if (obj == this) {
return true;

//Isn't this checking to see if the calling object is the same as the object we're passing to it?
Why doesn't this return false?
}
if (obj == null || obj.getClass() != this.getClass()) {
return false;
}

//How exactly do we check the values stored in each object though?
}