In the UK, both are awarded for studies undertaken after gaining a first (or ordinary) degree (eg BA, BSc).
They are also called post-graduate degrees and diplomas.
A graduate degree is a Master's or a Doctorate. A Master's requires at least 2 years of full-time study.
A graduate diploma requires 1 year of study, and can be used as an intermediate step along the way to a Master's degree.
(There are other diplomas which can be achieved without having any previous education.)