An area is a part of any surface. Areas are usually separated from everything around them by some kind of boundary, such as a fence or a line drawn around them on a piece of paper. Area also describes the total size of that area. Different shapes have different formula for calculating how big they are.
A field is a large area of ground that is used either to grow crops or for farm animals to graze on. Fields usually have a fence around the edge.
A paddock is a large open space, surrounded by a fence, that is used for keeping animals such as horses, cows and sheep in.