Since many other users have already explained and commented the storyline, I won't do it.
However, I'd like to restate that Bardem's interpretation is terrific, as also are those of the other actors and actresses in this film.
Reading the previous comments I've noticed that some people criticize the fact that the film doesn't show points of view opposed to euthanasia and that those little present are ridiculed. In my honest opinion this is far from true.
There are many characters that move in a gray zone between loving Ramón Sampedro and wanting him to stay, and understanding his desire to die. Most obvious of those are the family. For instance, Ramón's sister-in-law never talks for or against euthanasia. Another such character is Gené (the social rights activist) who, in the last moment, tells Ramón to re-think it all. The scene clearly shows that she doesn't want him to die.
Then there are characters who are clearly against euthanasia. Ramón's brother is clearly against it, as is his father ("There's only one thing worse than the death of son, and it's having a son that wants to die.") Other users have commented that the discussion between Ramón and the priest is ridiculed and filmed to make us think that Ramón is GOOD and the priest is BAD. Well, no doubt the scene is comic, but that doesn't mean the priest is caricatured or ridiculed. From my point of view, the comedy in this scene comes from the fact that the priest is trying to convince Ramón to keep on living using arguments totally alien to Ramón's thinking. The priest's speech goes on the line of "God gives and God takes", "We aren't the owners of our own lives, they belong to God"... and so on. The comedy arises from the fact that Ramón is atheist and all the priest is saying to him is therefore nonsense.
This film is the antithesis of manicheism, it leaves the spectator the chance to think on the subject and make up his/her own opinion. And above anything else is a chant of FREEDOM.