I group is a collection of users.
Each users has a primary group, and can have supplemental groups, that it belongs too. When you create a file with a user it is owned by that user and the gid of the file is set to the users primary group.
There are a bunch of utilities for changing the group, owner, and permissions of a file.
chgrp <group name> <file name> - Changes the owning group of the file.
chown <user name>:<group name-optional> <file name> - Changes the owning user and optionally the owning group of a file.
chmod <perms> <file name> - Changes the permissions associated with a file.
At a basic level users and groups are all done for the sake of access permissions to a file. Each file has permission for what the owner can do with them, what members of the owning group can do with them, and what everyone else can do with them. This way you can have a file that a subset of users can read, write, or execute without having to allow everyone on the system access to read, write, or execute it.
For more information check out the man pages for chmod, chgrp, and chown.
Also check out this article from tldp.org on File Access Permissions