Static members relate to a type not a concrete instance. So instance methods and variables concern the identity of the instance, e.g. an instance method can alter the state of the object.
To some extend this is a religious question. Some people prefer to create static methods in a class when they do not access any instance variable. Other people do not like this approach. Another usual usage is creating factory methods, able to create an instance of an object while itself being called statically.
Often there are utility classes that contain static methods to serve clients with some functionality that is not bound to any type context. But all too often these utility classes get abused by using them as a trash bin for all sorts of methods a programmer did not find a fitting class for. This may be a result of poor design.
A popular use of static variables is the definition of an logger instance (e.g. SLF4J) for a class because loggers are thread safe and can be used by all instances and static methods alike. One advantage is (see a comparison) that only one instance has to be created for all instances of the containing class.