Learn Italian: Ha il terrore che i topi tornino, perciò piange. - She is terrified that the mice will return, therefore she is crying.

The phrase ha il terrore che (she is terrified that) expresses a strong emotion or fear. When a sentence has two different subjects and the first clause expresses an emotion, attitude, or doubt followed by che, the second verb must be in the subjunctive mood.

The English translation says the mice "will return", so why aren't we using a future tense in Italian?

Italian does not have a future subjunctive tense. When the main verb expressing the emotion is in the present tense (ha), and the second action happens at the same time or in the future, you simply use the present subjunctive (tornino).

Why does tornino end in -ino instead of -ano?

The infinitive is tornare. A tricky part of the present subjunctive is that regular -are verbs flip their usual vowels. While the normal present indicative tense for "they return" is tornano, the subjunctive form becomes tornino.

What would happen if she were terrified of returning herself, rather than the mice returning?

You would not use the subjunctive. When the subject is the same for both actions (she is terrified and she is returning), Italian drops the che and uses di plus the infinitive verb instead: Ha il terrore di tornare.