Confidants are the Social Links of Persona 5 (if you've had experience with the previous entries in the series). They share in experiences with yours (as party members) or you bond with them over new experiences (et cetera) and are essential in breaking free from the chains that society has placed upon you. By interacting with them in your free time, you can increase that bond and unlock additional perks.
Party perks relate to what they can provide you in battle: Baton Pass (allows you to pass the turn off to someone else after Downing a foe with a Critical or Weakness attack), Harisen Recovery (chance of curing a status ailment of another party member upon their turn), ? Talk (chance of extending negotiations with that member in your active party), or special perks exclusive to them.
Et cetera perks relate to what each character can offer. Tae Takemi, the Death Arcana, can offer you additional medicines or accessories, eventually offering you cheaper prices too. Yuuki Mishima can help your group of thieves out with increases in experience.
To raise Confidant ranks, you must socialize with them and choose valid options that either assure their choices were correct or acceptable, or that you help them realize the mistake they are hiding behind. The theme of the game is freedom, so each Confidant has something that has them imprisoned within and ties to their social story.
Early on, you might want to socialize with some of the Confidants to get them started, but for me, the first one I maxed was Chariot before completing the second Palace.