It is not very clear why Capulet is having a party... we just know that there is one.
Romeo very reluctantly goes to the party. Benvolio is with Romeo when Capulet's servant who can't read asks Romeo to read the list of names that the servant needs to go invite. The servant acted as the local evite or text message... but had to use his feet and speak. His illiteracy forced him to rely on an apparently well-to-do looking person who happened to be a Montague. Romeo read the list and out of courtesy, the servant invited the boys as long as they weren't Montague.
Ironically, they were.
Benvolio encourages Romeo to go because Romeo was trippin over a girl named Rosaline and Ben thought if they went to the party, he could convince Romeo that other girls were prettier than Romeo's Rosaline.
Mercutio backed Benvolio up and further pressed Romeo to go even though Romeo didn't want to.
Juliet's relationship with her parents is good until she meets Romeo. After she falls for Romeo, Juliet's parents press her to marry Paris and they have no idea her heart is elsewhere. If she could be honest with them things would be different, but apparently she feels under these circumstances (like most teens) she needs to hide.
They appear to be having the party because it's traditional. We are not told exactly why, but Lord Capulet tells Paris (Act I, Scene 2) that it is something that they always celebrate.
Romeo is not really the one who decides to crash it. Benvolio and Mercutio are bugging him to go in. He doesn't want to because he's moping about Rosaline. But they keep making fun of him so he goes a long.
Juliet gets along fine with them. They care about her wishes (in terms of marriage) and she says that she will pay attention to their advice.