Hermione Granger (Muggle Born Witch)

Half Blood- Any wizard with half wizard and muggle ancestry or a wizard with non magical parents and at least one wizard grandparent.

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Harry Potter- Half Blood Wizard

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.

Ginny Weasley- Pure blood witch

Muggle: Non magic person  with non magic parents

Squib- A person born to magical parents and for some reason can't use magic.