So there are many ways of structuring objects (I'm talking of OOP here). For the question, I will use the classic "Car" example of OOP. Basically, How do I know when to make the car an object, or the wheel of a car an object, when both program structures would accomplish the goal?
How do I classify and categorize the parts of an object to determine whether or not they are better suited as simple attributes or variables of an object, or if they really need to be an object themselves?
Don't start out by classifying things - seems like people are too eager to start building inheritance hierarchies.
write down a list of specific, concrete scenarios - what your app will do, step by step. An object model is only useful if it does what you need it to do - so start working back from the scenarios to see what common objects and behaviours you can shake out of each one.
identify the "roles" in your scenarios - not necessarily actual class names - just vague "roles" that turn up when you think through concrete scenarios for how your software will work. These roles might later become classes, interfaces, abstract classes - whatever you need - at the start they're just placeholders for doing a type of work.
Work out what each role "does". The key is having a bunch of named roles - that identify things that the objects will do. Thins is about distilling out a set of things each role can do - they might do the whole thing, or put together a bunch of other objects to do the work, or they might co-ordinate the work... it depends on your scenarios.
The most important thing in OOD/OOP - is OBJECTS DO THINGS - not what's inside them - what they do.
Don't think about inheritance early on - because it will tie you up in overcomplicated hierarchies and make you think in terms of SQL-oriented programming rather than object-oriented programming. Inheritance is just one way of sharing common code. There are lots of other ways - delegation, mixins, prototype-based programming...
Here are some guidelines I came up with to help with this:
Well the first thing you have to realize is the OOAD ("Object-oriented analysis and design") is a tool and not a means to an end. What you get out of that process is a model, not a true representation of what you're modelling. That model makes certain assumptions. The purpose of that model is to solve a problem you have.
So how do you know how to design objects? How do you know if you've done it right? By the end result: has it solved your problem?
So, for the Car example, in some models a car count could simply be an integer count, for example the car traffic through an intersection in a traffic model. In such a model rarely do you care about the make, model or construction of cars, just the number. You might care about the type of vehicle to the point of is it a truck or car (for example). Do you model that as a Vehicle object with a type of Car or Truck? Or just separate carCount and truckCount tallies?
The short answer is: whichever works best.
The normal test for something being an object or not is does it have behaviour? Remember that ultimately objects = data + behaviour.
So you might say that cars have the following state:
Chances are you'll only care about a small subset of that: pick whatever is relevant. A racing game might go into more detail about the wheels, such as how hot they are, how worn, the width and tread type and so on. In such a case, a Wheel object could be said to be a collection of all that state (but little behaviour) because a Car has a number of Wheels and the Wheels are interchangeable.
So that brings up the second point about objects: an object can exist because of a relationship such that the object represents a complete set of data. So a Wheel could have tread, width, temperature and so on. You can't divide that up and say a Car has tread but no wheel width so it makes sense for Wheel to be an object since a Wheel in it's entirety is interchangeable.
But again, does that make sense for what're doing? That's the key question.
Attributes or variables are often "base" types of a language. The question is what you can sensibly abstract.
For example, you can reduce a Wheel to descriptors made up of base types like integers, floating-point values and strings, which represent characteristic attributes of any wheel: numberOfTreads, diameter, width, recommendedPressure, brand. Those attributes can all be expressed with base types to make a Wheel object.
Can you group some of those attributes into a more abstract arrangement that you can reuse, independent of a Wheel? I think so. Perhaps create a Dimensions object with the attributes diameter and width. Then your Wheel has a Dimensions object instance associated with it, instead of diameter and width. But you could think about using that Dimensions object with other objects, which may not necessarily be Wheel instances.
Going up the list, you can reduce a Car to be made up of base types, but also other objects, such as Wheel objects. It is sensible to do so, because other motor and non-motor vehicles (such as a Bicycle) also contain Wheel instances.
Abstracting Wheel and Dimensions lets you re-use these object types in different contexts you may not initially consider. It makes your life a little easier because you have less code to rewrite, in theory.
If you can create a hierarchy of objects, to the point where the deepest, lowest-level object is only made up of a few base types, that is probably a good place to start.
If it's true that "both program structures would accomplish the goal" equally well, then it doesn't matter which you pick.
If, however, the program does not have a single fixed "goal" but will evolve significantly over its lifetime, then pick either one for now, and refactor as necessary as future modifications dictate. We call it "software" for a reason.
I like Wirfs-Brock's Responsibility-Driven Design (RDD) and also recommend this updated (free paper) Responsibility-Driven Modeling approach by Alistair Cockburn.
In over 15 years of OO development, whenever I've felt I'm getting lost in a software architecture, going back to the RDD basics always helps me clarify what the software is supposed to be doing and how.
If you like a test-driven approach, this article shows how to relate RDD to mocking objects and tests.
There are some good answers here, but possibly more than you were looking for. To address your specific questions briefly:
How do I know when to make the car an object, or the wheel of a car an object, when both program structures would accomplish the goal?
When you need to distinguish one instance from another, then you need an object. The key distinction of an object is: it has identity.
Extending this answer slightly to classes, when the behaviors and/or properties of two similar objects diverge, you need a new class.
So, if you're modeling a traffic simulation that counts wheels, a Vehicle class with a NumberOfWheels property may be sufficient. If you're modeling a racing simulation with detailed road-surface and wheel-torque physics, each wheel probably needs to be an independent object.
How do I classify and categorize the parts of an object to determine whether or not they are better suited as simple attributes or variables of an object, or if they really need to be an object themselves?
The key distinctions are identity and behavior. A part with unique existence is an object. A part with autonomous behavior requires its own class.
For example, if you're creating a very simple car-crash simulation, NumberOfPassengers and DamageResistance may be sufficient properties of a generic Vehicle class. This would be enough to tell you if the car was totalled and the passengers survived. If your simulation is much more detailed, perhaps you want to know how far each passenger was thrown in a head-on collision, then you would need a Passenger class and distinct Passenger objects in each Vehicle.
Grow your classes bottom-up.
1) Class boundaries and semantics depend on context. Until you have a context, you don't have anything. (You may not even have a car in your example). Context is given by the user story (or use case).
2) Throw all the state and behavior suggested by the given context into one class (you could name this after the user story if you would like).
3) Use systematic Refactoring to tease this class apart into separate classes. While refactoring, use existing classes as reuse opportunities.
When you're done, you'll have a set of well-defined classes that are just enough to fulfill the needs of the given user story (and the user stories that came before).