mündlich
oral
Done through speaking; not written, but expressed or communicated by mouth.
Die Prüfung besteht aus einem schriftlichen und einem mündlichen Teil.
The exam consists of a written and an oral part.

oral / verbal
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oral
Done through speaking; not written, but expressed or communicated by mouth.
Die Prüfung besteht aus einem schriftlichen und einem mündlichen Teil.
The exam consists of a written and an oral part.

The word 'mündlich' (oral) derives directly from 'der Mund' (the mouth). The ending '-lich' turns the noun into an adjective and means something like 'in the manner of a mouth,' so spoken.
Wir haben eine mündliche Abmachung getroffen.
We made a verbal agreement.
'Mündlich' (oral) is very often used in fixed combinations, especially in education and law. Typical examples are 'mündliche Prüfung' (oral exam), 'mündliche Zusage' (verbal promise/assurance), 'mündliche Vereinbarung' (verbal agreement), or 'mündlicher Vertrag' (oral contract).
Eine mündliche Zusage ist zwar nett, aber ich hätte lieber etwas Schriftliches.
A verbal promise is nice, but I would prefer something in writing.
'Mündlich' (orally) can also be used as an adverb to describe a verb. It then answers the question 'How?'. It often appears with verbs of communication.
Bitte teilen Sie mir Ihre Entscheidung mündlich mit.
Please inform me of your decision orally.
personal / in person
'persönlich' and 'mündlich' can both be important in conversations and when giving information. 'persönlich' describes that someone is there themselves or handles something themselves, often directly on site. 'mündlich' describes that something is communicated orally and not in writing. The words are easy to confuse because an in-person conversation is often also oral. But 'mündlich' can also be by phone or in a voice message. 'persönlich' emphasizes direct presence; 'mündlich' emphasizes the spoken form.

Die Zusage war nur mündlich, deshalb bittet Tom um eine schriftliche Bestätigung.
The acceptance was only verbal, so Tom asks for written confirmation.
Im Rathaus muss man den Antrag persönlich abgeben.
At city hall, you have to submit the application in person.