
abweisen
to reject
To reject or refuse a person, a request, an offer, or a suggestion.
Sie hat sein Angebot, ihr zu helfen, kühl abgewiesen.
She coolly rejected his offer to help her.
to reject / turn away
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to reject
To reject or refuse a person, a request, an offer, or a suggestion.
Sie hat sein Angebot, ihr zu helfen, kühl abgewiesen.
She coolly rejected his offer to help her.

to turn away
To deny someone entry or admission; to send someone away.
Der Türsteher hat mehrere Gäste abgewiesen, weil der Club voll war.
The bouncer turned away several guests because the club was full.

to dismiss
In a legal context, to reject a lawsuit, motion, or claim as unfounded or inadmissible.
Das Gericht hat die Klage mangels Beweisen abgewiesen.
The court dismissed the lawsuit due to lack of evidence.

to repel
A property of materials: not allowing certain substances such as water or dirt to penetrate.
Die neue Jacke ist aus einem Material, das Wasser und Schmutz abweist.
The new jacket is made of a material that repels water and dirt.
'Abweisen' is a separable verb. The prefix 'ab-' stands at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Der Chef wies meinen Vorschlag sofort ab.
The boss rejected my suggestion immediately.
The verb is made up of the prefix 'ab-' (away) and the verb 'weisen' (to show, to guide). The literal meaning 'to point away' or 'to send away' is the basis for all meanings of the verb.
Der Hund wurde am Eingang des Supermarktes abgewiesen.
The dog was turned away at the entrance to the supermarket.
You can reject people ('einen Verehrer abweisen' — to reject an admirer) or their offers. Typical objects are: 'ein Angebot' (an offer), 'eine Bitte' (a request), 'ein Kompliment' (a compliment), or 'ein Vorschlag' (a suggestion).
Er hat ihr Jobangebot höflich, aber bestimmt abgewiesen.
He rejected her job offer politely but firmly.
In this meaning, 'abweisen' (to dismiss/reject) is used almost exclusively in a legal context. It is often used in the passive to report what happened to a lawsuit or motion.
Der Antrag wurde vom Gericht als unbegründet abgewiesen.
The motion was dismissed by the court as unfounded.
For materials, the adjective form 'abweisend' is very common. People speak of 'wasserabweisenden' Jacken (water-repellent jackets) or 'schmutzabweisenden' Oberflächen (dirt-repellent surfaces).
Diese Hose ist aus einem schmutzabweisenden Stoff gemacht.
These pants are made of a dirt-repellent fabric.
to reject
Both verbs can be used to reject a request or proposal. Ablehnen focuses more on the decision against the request itself; abweisen can additionally emphasize the person or their attempt and can therefore sound more personal. With ablehnen, focus mainly on the request; with abweisen, you may instead focus on the person who was turned away.

Die Behörde wies den Antragsteller mit seiner Bitte ab.
The authority turned the applicant away when he made his request.
Die Behörde lehnte seinen Antrag auf Unterstützung ab.
The authority rejected his application for support.