austeilen
to hand out
To distribute something to a group of people so that everyone gets a share.
Der Lehrer teilte die Arbeitsblätter an die Schüler aus.
The teacher handed out the worksheets to the students.

to hand out / dish out
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to hand out
To distribute something to a group of people so that everyone gets a share.
Der Lehrer teilte die Arbeitsblätter an die Schüler aus.
The teacher handed out the worksheets to the students.

'Austeilen' is a separable verb. The prefix 'aus-' goes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Die Freiwilligen teilen das Essen aus.
The volunteers hand out the food.
The thing being distributed is in the accusative. The people who receive something are often connected with the preposition 'an' + accusative.
Die Lehrerin teilt die Zeugnisse an die Schüler aus.
The teacher hands out the report cards to the students.
In the literal meaning, 'austeilen' is often combined with things that are distributed to a group. Typical examples are: 'Getränke austeilen' (to hand out drinks), 'Karten austeilen' (to hand out cards), or 'Werbung austeilen' (to hand out advertising flyers).
Vor dem Spiel teilte der Trainer die Trikots aus.
Before the game, the coach handed out the jerseys.
In the figurative meaning, the expression 'austeilen, aber nicht einstecken können' (to dish it out but not be able to take it) is very common. It describes a person who likes to criticize others but cannot tolerate criticism themselves.
Mein Kollege kritisiert ständig alle, aber wehe, man sagt etwas zu ihm. Er kann gut austeilen, aber nicht einstecken.
My colleague constantly criticizes everyone, but just try saying something to him. He can dish it out, but he can't take it.
to distribute / spread
'verteilen' and 'austeilen' can be used very similarly when a group of people receives something. Both fit, for example, with sheets of paper, food, cards, or gifts. 'austeilen' often emphasizes that everyone in the group gets a share. 'verteilen' is a little more general and can also mean that things are brought to different places. In many everyday situations involving a group, however, you can use both words.

In der Pause hat Mehmet die Hefte an die Klasse ausgeteilt.
During the break, Mehmet handed out the notebooks to the class.
Morgen verteilt der Verein warme Kleidung an die Besucher.
Tomorrow the association will distribute warm clothing to the visitors.