Half Blood- Any wizard with half wizard and muggle ancestry or a wizard with non magical parents and at least one wizard grandparent.
In some cases half bloods could be the off spring off two muggleborns.
Muggleborn- A Wizard born to muggle parents
The reason they have magic is because they are descended from a squib that married a non magic person and then the magic comes back genertions later.
Pureblood- A Person of full magical Ancestry.
Muggle: Non magic person with non magic parents
Squib- A person born to magical parents and for some reason can't use magic.