Think of a harddrive as a house. If you move into a house that's freshly built, you'll see brickwork, and it'll be cold. Also, there's so much space, but the uneven floor will make it difficult to get anything in.
Partitioning is like adding floors and walls in the house. This allows rooms to be made, but how things are stored are all over the place.
Formatting prepares the drive for use by decorating the rooms in a way stuff can be stored nicely.
You partition the harddrive to tell the computer where the limits of the disk are, and then format it to allow the operating system (eg Windows, and there may be more than one system installed, but that's advanced topics) to know how things are stored on the drive.
Does this make sense? I've been telling people this for years, and so I hope it does!