You find a person that you like. You probably like that person more than other people, maybe more than everyone else you know. Maybe you share some goals or beliefs, some hobbies. whatever, but for some reason you get on together pretty well. You feel inclined to treat that person specially, giving them more of your attention and making them a higher priority than others. They may make you laugh and you may think they are sexy, too. Maybe you really really like sex with that person, or you really like other things you do together, or both. You talk to each other about your hopes and dreams and you work out your problems and quirks together, because it seems like how you should do it. There are often feelings of mushy gushy warm and fuzzy inside, sometimes jealousy, and for the most part all you inexplicably want is to make that person happy. You feel safe with each other, you trust each other, you tell each other about that time you ate a bug but you were 12 man, wtf or that time you let the boy next door pee in your hair and never told anyone. Some people think that the other person is the best possible choice when thinking about starting a family. You hold her hair back when she gets sick and you eat dinner at her mom's every Sunday even though it's not fun because you want to do things for her. You help him save up for that new gadget and you struggle to buy him just the right birthday present every year, because this is more or less what love is.
There are lots of motivations for being a relationship. This is basically what it is.
Also what nobo said.