wieder
again
Expresses repetition; one more time, again.
Nächstes Jahr fahren wir wieder nach Italien.
Next year we're going to Italy again.

again / back
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again
Expresses repetition; one more time, again.
Nächstes Jahr fahren wir wieder nach Italien.
Next year we're going to Italy again.

The meaning of 'wieder' (again/back) depends on the context. In connection with a simple action, it usually means 'one more time'. In connection with a state or place, it often means 'back'.
Kannst du das bitte wieder sagen? (noch einmal)
Can you please say that again? (one more time)
'Wieder' (again/back) is often a prefix for separable verbs that express a return or restoration. Examples include 'wiederkommen' (to come back), 'wiederfinden' (to find again), and 'wiedersehen' (to see again).
Wann kommst du wieder nach Deutschland?
When are you coming back to Germany?
The combination 'immer wieder' (again and again) expresses a frequent, repeated action. It emphasizes that something happens not just once, but multiple times and regularly.
Obwohl ich es ihm verbiete, macht er immer wieder den gleichen Fehler.
Although I forbid him to do it, he keeps making the same mistake again and again.
back
'wieder' and 'zurück' can both have to do with returning. 'zurück' usually emphasizes the direction toward the starting point or to a previous place. 'wieder' emphasizes that a previous state is present again or that something is happening again. You can say: 'Ich gehe zurück nach Hause' (I’m going back home) for the movement. 'Ich bin wieder zu Hause' (I’m back home / I’m home again) says more that the state is now as it was before. They can be confused because both can fit with returning, but 'zurück' shows direction, while 'wieder' shows repetition or the old state.

Nach zwei Wochen Urlaub bin ich wieder im Büro.
After two weeks of vacation, I’m back in the office.
Er geht zurück zum Supermarkt, weil er seine Tasche vergessen hat.
He is going back to the supermarket because he forgot his bag.