der Italiener
Italian
A male person who comes from Italy or has Italian citizenship.
Mein neuer Kollege ist Italiener und kommt aus Rom.
My new colleague is Italian and comes from Rome.

Italian / Italian restaurant
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Italian
A male person who comes from Italy or has Italian citizenship.
Mein neuer Kollege ist Italiener und kommt aus Rom.
My new colleague is Italian and comes from Rome.

The word 'Italiener' (Italian) is formed from the country name 'Italien' (Italy) with the ending '-er'. This ending is often used to refer to male inhabitants of countries or cities.
Ein Mann aus Berlin ist ein Berliner, ein Mann aus Italien ist ein Italiener.
A man from Berlin is a Berliner; a man from Italy is an Italian.
You can tell from context whether a person or a restaurant is meant. If it's about food or going out, the restaurant is usually meant, often in combination with prepositions like 'zum' (to the) or 'beim' (at the).
Lass uns heute Abend zum Italiener gehen.
Let's go to the Italian restaurant tonight.
When 'der Italiener' (the Italian restaurant) means a restaurant, it is often used with certain phrases: 'zum Italiener gehen' (to go to the Italian restaurant), 'beim Italiener essen' (to eat at the Italian restaurant), or 'einen Tisch beim Italiener reservieren' (to reserve a table at the Italian restaurant).
Wir haben für Samstag einen Tisch beim Italiener reserviert.
We reserved a table at the Italian restaurant for Saturday.
The term 'der Italiener' (the Italian restaurant) for an Italian restaurant is very common, but it is considered colloquial.
Ich kenne einen guten Italiener in der Nähe.
I know a good Italian restaurant nearby.
Italian
'italienisch' and 'der Italiener' both have to do with Italy, but grammatically they do not belong to the same group. 'italienisch' is an adjective and describes things like food, music, language, or culture. 'der Italiener' is a noun and usually means a man from Italy. Colloquially, 'zum Italiener gehen' (to go to the Italian restaurant) can also mean going to an Italian restaurant. The words can be confused when talking about origin or language: a person can be Italiener, but a language or a dish is italienisch.

Im Supermarkt spricht er mit einem Italiener über frische Tomaten.
In the supermarket, he talks with an Italian man about fresh tomatoes.
Frau Wolf kocht heute ein italienisches Gericht.
Ms. Wolf is cooking an Italian dish today.