If there are two neighbors, and neighbor A has an apple tree, and a branch of the apple tree hangs on the side of neighbor B's garden, is neighbor B allowed to take an apple from this branch, because it is on his side of the garden? Or is it considered theft?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

The basic principle is that it is impermissible to take the wealth of a Muslim except with his consent. Allah, the Exalted, says (what means): {And do not consume one another's wealth unjustly ...} [Quran 2:188] Moreover, the Prophet, sallallaahu ‘alayhi wa sallam, said, "The property of a Muslim is unlawful to others except if he gives it willingly." [Al-Albaani graded it saheeh (sound)]

Hence, it is prohibited to take the fruits dangling from the neighbor's tree except with his permission. Please refer to fatwa 90231.

Allah knows best.