What is the difference between class and instance methods?

All the technical details have been nicely covered in the other answers. I just want to share a simple analogy that I think nicely illustrates the difference between a class and an instance:

A class is like the blueprint of a house: You only have one blueprint and (usually) you can't do that much with the blueprint alone.

An instance (or an object) is the actual house that you build based on the blueprint: You can build lots of houses from the same blueprint. You can then paint the walls a different color in each of the houses, just as you can independently change the properties of each instance of a class without affecting the other instances.

Like most of the other answers have said, instance methods use an instance of a class, whereas a class method can be used with just the class name. In Objective-C they are defined thusly:

@interface MyClass : NSObject

+ (void)aClassMethod;
- (void)anInstanceMethod;

@end

They could then be used like so:

[MyClass aClassMethod];

MyClass *object = [[MyClass alloc] init];
[object anInstanceMethod];

Some real world examples of class methods are the convenience methods on many Foundation classes like NSString's +stringWithFormat: or NSArray's +arrayWithArray:. An instance method would be NSArray's -count method.

Like the other answers have said, instance methods operate on an object and has access to its instance variables, while a class method operates on a class as a whole and has no access to a particular instance's variables (unless you pass the instance in as a parameter).

A good example of an class method is a counter-type method, which returns the total number of instances of a class. Class methods start with a +, while instance ones start with an -. For example:

static int numberOfPeople = 0;

@interface MNPerson : NSObject {
     int age;  //instance variable
}

+ (int)population; //class method. Returns how many people have been made.
- (id)init; //instance. Constructs object, increments numberOfPeople by one.
- (int)age; //instance. returns the person age
@end

@implementation MNPerson
- (id)init{
    if (self = [super init]){
          numberOfPeople++;
          age = 0;
    }    
    return self;
}

+ (int)population{ 
     return numberOfPeople;
}

- (int)age{
     return age;
}

@end

main.m:

MNPerson *micmoo = [[MNPerson alloc] init];
MNPerson *jon = [[MNPerson alloc] init];
NSLog(@"Age: %d",[micmoo age]);
NSLog(@"%Number Of people: %d",[MNPerson population]);

Output: Age: 0 Number Of people: 2

Another example is if you have a method that you want the user to be able to call, sometimes its good to make that a class method. For example, if you have a class called MathFunctions, you can do this:

+ (int)square:(int)num{ 
      return num * num;
}

So then the user would call:

[MathFunctions square:34];

without ever having to instantiate the class!

You can also use class functions for returning autoreleased objects, like NSArray's

+ (NSArray *)arrayWithObject:(id)object

That takes an object, puts it in an array, and returns an autoreleased version of the array that doesn't have to be memory managed, great for temperorary arrays and what not.

I hope you now understand when and/or why you should use class methods!!