While Vim doesn't explicitly contain classes, dictionaries support having methods on them, supporting the object-oriented programming paradigm. There are two ways to define a method on a dictionary.

Given an existing dictionary, animal_names:

let animal_names = {
   'cat': 'Miss Cattington',
   'dog': 'Mr Dogson',
   'parrot': 'Polly'
   }

You could do the following to add a method to it:

function animal_names.GetGreeting(animal)
  return self[a:animal] . ' says hello'
endfunction

You can now execute the function:

:echo animal_names.GetGreeting('cat')
Miss Cattington says hello

You can use self (just like in Python!) to refer to dictionary keys.

In the previous example, GetGreeting becomes a callable dictionary key. Effectively, animal_names becomes this:

{
   'cat': 'Miss Cattington',
   'dog': 'Mr Dogson',
   'parrot': 'Polly',
   'GetGreeting': function <...>
   }

For the next example, let's wrap our animal_names dictionary in a slightly more generic animals. This will make our next example behave slightly more like classes we're used to in other languages (and would help us avoid name collisions):

let animals = {
   'animal_names' : {
     'cat': 'Miss Cattington',
     'dog': 'Mr Dogson',
     'parrot': 'Polly'
     }
   }

You can also define a function before having a dictionary ready, by using dict keyword after the function name:

function GetGreeting(animal) dict
  return self.animal_names[a:animal] . ' says hello'
endfunction

Now, we'll need to assign the function as another dictionary key:

let animals['GetGreeting'] = function('GetGreeting')

Now, you can call GetGreeting in the same manner:

:echo animals.GetGreeting('dog')
 Mr Dogson says hello