heir vs heir-at-law

heir and heir-at-law both are nouns.

Word	Noun	Adjective	Verb	Adverb
heir	Yes	No	No	No
heir-at-law	Yes	No	No	No

As nouns, heir-at-law is a hyponym of heir; that is, heir-at-law is a word with a more specific, narrower meaning than heir:

heir: a person who is entitled by law or by the terms of a will to inherit the estate of another
heir-at-law: the person legally entitled to inherit the property of someone who dies intestate

Other hyponyms of heir include heir apparent, heiress, inheritress, inheritrix, heir presumptive.

heir (noun)	heir-at-law (noun)
a person who is entitled by law or by the terms of a will to inherit the estate of another	the person legally entitled to inherit the property of someone who dies intestate
a person who inherits some title or office