The verb plaire is most often used in the expressions s'il vous plaît (formal) and s'il te plaît (informal), which, as you probably know, both mean "please"––or more accurately, "if it pleases you." "To please" is the basic meaning of plaire:
Ça peut pas leur plaire.
That can't please them.
Cap. 18, Le Journal - Yann Arthus Bertrand
Another way of saying "to please" is faire plaisir (literally, "to make pleasure"):
Je sais que ça va pas te faire plaisir
I know this isn't going to please you
Cap. 18, Le Jour où tout a basculé - Mon père s'oppose à ma passion - Part 7
If something pleases you, that means you like it. Indeed, plaire can also mean "to like" or "enjoy":
Une autre œuvre qui me plaît beaucoup
Another work that I like a lot
Cap. 35, Patrice Zana - L'artiste et ses inspirations - Part 2
OK, je te plais pas.
OK, you don't like me.
Cap. 52, Le Jour où tout a basculé - À la recherche de mon père - Part 4
Ce livre plaît à tout le monde.
Everyone enjoys that book.
We could certainly translate the above examples as "another work that pleases/appeals to me a lot," "OK, I'm not pleasing/appealing to you," and "that book is pleasing/appealing to everyone." But plaire is used a bit more generally than "to please," so you'll usually see it translated as "to like" or "enjoy" with the subject and object inverted (ce livre plaît à tout le monde = everyone enjoys that book). Note that plaire always takes an indirect object (plaire à quelqu'un, "to please/be pleasing to someone").
When plaire is reflexive (se plaire, literally "to please oneself"), it means "to be happy" or "to enjoy being somewhere":
Est-ce que tu t'y plais?
Are you enjoying yourself here?
Cap. 4, Yabla à Nancy - Université Nancy 2
Elles se plaisent à Lindre
They like Lindre
Cap. 21, Lionel - à Lindre-Basse - Part 6
Or, in the plural, it can mean "to like one another," "to enjoy each other's company":
Ils se sont plu immédiatement.
They liked each other instantly.
And for life's unpleasant moments, there's the verb déplaire (to dislike, displease, irritate, upset):
Ses plaisanteries déplaisent à ma mère.
My mother doesn't like his jokes. (His jokes irritate my mother.)
There's also the expression n'en déplaise à (with all due respect to, with apologies to, no offense to):
Pas de fiole de cyanure, n'en déplaise à Shakespeare
No vial of cyanide, no offense to Shakespeare
Cap. 47, Grand Corps Malade - Roméo kiffe Juliette
We hope you're pleased with this lesson on plaire!