What does "obey" mean in marriage vows?

It means the vows were written many generations ago when women were not considered a man's equal in any way. They ran a man's house, they bore a man's children, and that was their "place" in society. Most marriages now don't include that word in their vows. Not if they are saying vows they mean to keep!

It means you respect your partner to the point where you try not to disappoint them deliberately. Many people overreact to this word.

it means the husband and wife should respect each other's  feelings for example if the husband tells the wife he wants to go to the club and wife says I don't want you to go to the club cause there are to many slutty women there then the husband must respect her wishes and not go also the same goes for women as well if her husband tells her not to go hangout with her single friends cause he doesn't want her single friends influencing her to do what single people then the wife must respect her husband wishes and not hangout with her single friends

It comes from a traditional role.  If a decision is essential someone has to have the authority to make it.  Traditionally the male was given a greater right to make a decision and the wife had to abide by that.  So obey means giving way to the husband where it is necessary.  SOMEONE has to give way.

Google it. There's all sorts of information there.