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I find it helps to think of a given problem in terms of 'things'.
If the problem can be thought of as having one or more 'things', where each 'thing' has a number of attributes or pieces of information that refer to its state, and a number of operations that can be performed on it - then OOP is probably the way to go!
OO allows for logic related to an object to be placed within a single place (the class, or object) so that it can be decoupled and easier to debug and maintain.
What I have observed, is that every app is a combination of OO and procedural code, where the procedural code is the glue that binds all your objects together (at the very least, the code in your main function). The more you can turn your procedural code into OO, the easier it will be to maintain yor code.
It depends by the problem: the OOP paradigm is useful in designing distribuited systems or framework with a lot of entity living during the actions of the user (example: web application).
But if you have a math problem you will prefer a functional language (LISP); for a performance-critical systems you will use ADA or C, etc etc.
The language OOP is useful because too it use probabily the garbage collector (automatic use of memory) in the run of program: you you program in C a lot of time you must debug and correct manually a problem of memory.
OOP is useful when you have things. A socket, a button, a file. If you end a class in er it is almost always a function that is pretending to be a class. TestRunner more than likely should be a function that runs tests(and probably named run tests).