Durante la manutenzione l’ascensore resterà fuori servizio per due ore.

durante is a preposition meaning “during” or “throughout,” and it must be followed by a noun (here la manutenzione).
By contrast, mentre is a conjunction meaning “while” and is followed by a verb clause. For example:

• “Mentre l’ascensore è in manutenzione, prendete le scale.”
  You cannot say mentre la manutenzione because mentre cannot introduce a bare noun.

restare literally means “to remain” or “to stay” and focuses on the continuation of a state.
You could substitute essere without changing the basic meaning:

• “L’ascensore sarà fuori servizio per due ore.”
  But notices often use restare to signal “will remain out of service,” stressing its ongoing condition.

resterà is the third-person singular of the futuro semplice of restare.
Formation: take the infinitive stem rest- and add the future ending -erà.
Full conjugation (singular):

• io resterò
• tu resterai
• lui/lei resterà

fuori servizio is an idiomatic adjective phrase meaning “out of service” or “out of order.”

• fuori (“outside”) functions like an adverb here, modifying servizio (“service”).
  Together they form a fixed expression used for machines, elevators, etc., that aren’t operational.