Relationships are the "alliance status" of you and other AI tribes. If you look on the stats menu in game, you will see you and the other tribes you are facing, if you have discovered them. If there is nothing around their icons, you are neutral with them. They have a 50/50 chance of attacking you or leaving you alone. A halo around their heads means you are allies, and they have a decreased likelihood of attacking. Red arrows pointing to their head means they HATE you and will kill you at all costs. This happens if you attack them, attack their allies, destroy their allies, or capture their cities. They will become your allies if you attack their enemies or meet them in easy mode.
If you are more powerful than another tribe upon contact (better leveled cities, more points), there is a high chance of them being allied with you, especially in easy mode.
If you are in Crazy difficulty, no matter your standing, there is an extremely high chance of them being instantly hostile.