Is a monoid the same thing as a group but without the requirement for elements to have an inverse element?
Essentially you're correct, but I wouldn't use the expression "same thing." Every group is a monoid, but not every monoid is a group. The integers under multiplication is an example of an abelian monoid which isn't a group (as are the rationals and reals).
A monoid has less structure than a group, so more structures are monoids, but monoids (that aren't groups) don't behave as nicely as groups.
Another useful example of a monoid is the set of all (partial) functions from A to A. There is always an identity function, and some functions are bijections so have inverses. In general though, your functions will not have inverses. However, injections and surjections will have left or right inverses.