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 can happen for a variety of reasons (i.e. you attack/encroach on them, they become more powerful than you). 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.
There are 4 different kinds of greetings when meeting a tribe. Each greeting represents different kinds of tribal relationships, and will determine your relationship with a new tribe.
"They bow in awe of your mighty civilization." This means they are peaceful with you and won't attack you, since you have more points. To get this greeting you must be significantly stronger than your opponent.
"They salute you friendly but pay no attention to your tiny realm." This means they are at peace with you, and they don't find you strong enough to be worth bothering with. To get this greeting you must be much weaker than them.
"They seem a bit hostile and greet you suspiciously." This means they see you as a threat, and there is a high likelihood that they will attack you soon. To get this greeting, you must be about the same strength as them.
"They laugh at your puny excuse of a tribe." This means they are actively at war with you and will actively target you. To get this greeting they must have a large advantage over you.
If you are in Crazy difficulty, no matter your standing, there is an extremely high chance of them being instantly hostile.
There are always ways to change your standing with an opponent. If you want to become friends with another tribe who is currently either hostile or neutral, there are 2 ways to do so.
The first method is to not attack them. If they attack your units, do not attack their units back. Usually, if you are more powerful than them, they eventually cease being hostile or neutral and become allies again. However, if you have provoked the enemy tribe too much or if the enemy is much more powerful than you, they will not stop attacking you. This method rarely ever works in hard mode, and even less in crazy mode.
The second and more effective method is to wait for another tribe to attack the tribe you want to change your standing with. Once they are attacked, you must help them defend without sieging their cities. If you do move your units into their cities, even without capturing them, there is a high chance that they will become hostile.
Learn more about it here.