1. Out can indicate removal.

Pattern: verb + noun + out
Please take the trash out.

Typical verbs used with out:
carry, cross, cut, get, kick, leave, move, take, tear, throw

2. Out can indicate distribution.

Pattern: verb + noun + out
The teacher told me to hand these papers out.

Typical verbs used with out:
give, hand, mail, pass, send

3. Out of indicates movement from inside.

Pattern 1: verb + out of + noun
He was freezing when he got out of the water.

Typical verbs:
come, crawl, drink, drive, eat, fall, get, go, jump, hop, run, step

Pattern 2: verb + noun + out of + noun
She took the cake out of the oven.

Typical verbs:
drive, get, grab, move, pour, pull, push, rip, sip, squeeze, take, tear

4. Out of indicates absence.

Pattern: be + out of + noun
The boss is out of the office.
My neighbors are out of the country this month.

Expression:
to be out of town—to be absent from one's place of residence
The boss is out of town this week.

5. Out of indicates a distance from.

Pattern: verb + out of + noun of place
The restaurant is about three miles out of town.
They live two blocks out of the city limits.

6. Out of can mean no longer in supply.

Pattern: be/run + out of + plural or noncount noun
I can't make a cake because I am out of eggs.
They had to walk to the gas station because they ran out of gas.

Typical noncount nouns used after out of:
breath, gas, luck, money, stock, time, work