Tifa Lockhart
The original game hinted at competition between the two for Cloud's affection - although they always demonstrated respect for each other. They were both polite, cooperative, and focused on the mission to stop Sephiroth. This dynamic is noticably different in the remake projects. They share a warmer, sisterly bond - often in cahoots at Cloud's expense. In real life, the two would have likely bonded over the loss of their parents. In both the original and remake, Tifa's clearly displays emotional devastation in reaction to Aerith's death.
Yuffie Kisaragi
Aerith and Yuffie have a warm, little sis/big sis dynamic. The original game shows minimal interaction, but in the newer entries their relationship is friendlier and light-hearted. Yuffie completely drops her tough guy act and openly sobs after Aerith dies.
Barret Wallace
They share a cooperative, teamate-like bond. Barret quickly develops respect for Aerith as she shows care for Barret's daughter, Marlene, during the platefall crisis. When Aerith dies, Barrett reacts with a mix of grief and rage.
Red XIII
Both Red XIII and Aerith both share a spiritual connection to the planet. They support each other through highs and lows, forming a close bond. Their relationship is rooted in empathy, due to both having been subjected to the whims of Hojo in the Shinra labs. At one point, Hojo planned to breed them, although this is treated as a scientifically unethical and disturbing concept.
Vincent Valentine
Although comrades, Aerith and Vincent don't really display any kind of significant bond. He treats Aerith with respect, as he does any other member of the group.