Utility class can be any class which contain methods which might be used frequently in a lot of class in the SFDC instance. eg. you would need to get the name of the record type from the id. So you write a generic method in the utility class which does the job for you and you can reuse the code of that class.
You can just write the class and methods, then reference it in other Apex classes.
class MyUtil{
public void myMethod(){
}
}
trigger myTrigger{
MyUtil mu = new MyUtil();
mu.myMethod();
}