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. This is usually found when meeting a tribe in moderate difficulties with around your size, when a tribe feels threatened by your advance, or when you have just saved their city. In the latter case, they will likely become friends next turn.
A halo around their heads means you are friends with them, and they have a low likelihood of attacking you unless you are the last two tribes remaining, and may occasionally trust you enough to leave their cities open. However, this is very rare.
Red arrows pointing to their head means they are enemies 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 actively try to attack and conquer your tribe, and is nearly impossible to change unless you save their city from enemy tribes or fight a common foe whilst not attacking them.
If you are more powerful than another tribe upon contact (better leveled cities, more points), there is a high chance of them being friends with you, especially in easy mode.
Relationships also depend on what difficulty you’re playing. On Easy difficulty, every tribe you meet is never at war with you unless provoked. On Normal and Hard difficulties, relations vary by your power, if you’re at war with their enemies, or how many tribes you eliminate. On Crazy difficulty, total war is almost always the case. It is extremely hard to be friends with another tribe and killing tribes might cost you relations.
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, or you share a common enemy.
"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 will be neutral, but 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, or be a Crazy difficulty AI.
If you are in Crazy difficulty, no matter your standing, there is an extremely high chance of them being instantly hostile.
If in Easy difficulty, every tribe you meet will be at peace with you. The likelihood of them attacking you unprovoked is low or never.
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.