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It is a game designer's job to design.
That might seem like a tautology, and it is, so let me elaborate.
A game designer is responsible for the game's storyline and gameplay style.
There are different types of designers, such as those who specialise in the story or in mechanics, but there are also more and more designers who are a Jack of all trades, they do a bit of everything.
A designer will often see a project from start to finish, from designing the initial concept all the way through to production and user testing.
A team of designers might work on a project, but on smaller projects, one lead designer is often enough.
It is the designer's job to create a coherent play experience from the overarching storyline to how responsive the buttons should be.
They also oversee the work of other game professionals, such as animators, to make sure their work fits with the overall feel of the game.
Animator
Much in the way that designers design, animators animate.
Let's start with a question, what is an animation?
The animation is the process of taking a static object and an idea of how it should move, and applying techniques to make it move in the desired way.
This is what animators do; from making characters walk and talk to making sure the hands on a clock move correctly.
Animators play an important part in the games pipeline, making sure that the game's world behaves the way the player expects. A designer will oversee this process, to make sure those laws of interaction make sense within the concept of the game.
Career Paths
The career progression path is very similar for both careers; you would begin as a junior, working as part of a team (of variable size depending on the size of the studio), and progress to senior level when you have the required experience.
You may even become experienced enough to become a project lead, taking control of the entire project and overseeing all the processes.
It is important to note that these careers are parallel but not the same; both are needed for a successful game.
Educational Requirements
A game designer's education would be most likely built on an understanding of game mechanics and the way that the player interacts with the game.
This often takes the form of a college education and can include classes in design methodologies.