Your tour of the rooms begins with the Room of the Signatura. Originally this room was the pope's private library and it was divided into four humanities subjects: theology, philosophy, poetry, and law. There were large frescoes above the bookshelves that depicted the subject's most important facts and people.
After careful graphic preparation, Raphael first painted the fresco entitled Disputation of the Holy Sacrament on the wall reserved for the theology texts. At the center of the composition you can see an altar with the consecrated host. Around it is a group of clergy discussing the mystical nature of the Eucharist. Raphael depicted saints and theologians in a double semicircle: some are on the ground and others are seated in the clouds around the figures of God the Father and Jesus Christ.
Now go to the next fresco called The School of Athens, which was frescoed in the philosophy section. Raphael shows all the greatest thinkers of antiquity as if they were together on a theatrical stage dominated by a gigantic building, which was perhaps inspired by the works on Saint Peter's new Basilica. The philosophers are separated into groups: in the center you can see Plato and Aristotle making gestures that synthesize their doctrines. Have fun looking at how Raphael portrayed many of his illustrious colleagues: it's easy to recognize the face of Leonardo da Vinci in Plato, who is indicating the sky, the sulky Heraclitus in the center resting his cheek in his hand has Michelangelo's features, the man tracing geometric figures with the compass is the mathematician Euclid with the face of Bramante, and at the far right you can also see Raphael himself.
Now move on to the fresco entitled The Parnassus.
The idea of continuity between classical antiquity and the present is depicted in the scene of Mount Parnassus, which was on the wall with poetry texts. Apollo plays a lyre in the center while he is surrounded by the nine best muses and poets of all time (Homer, Saffo, Virgil, Dante, Petrarch, Ariosto...); you can recognize them because they are all crowned with laurels.
Lastly on the wall for the texts about law, you can admire the fresco entitled The Cardinal Virtues, which are Strength, Prudence, and Temperance. On the ceiling above them you can see the tondo painting with Justice. Raphael didn't have enough time to paint the two scenes below, as the pope wanted him to immediately move on to the decoration of the next room.
FUN FACT: as I mentioned, Michelangelo was depicted as the philosopher Heraclitus, but you may have noticed that he's wearing boots in the foreground. This is not a random detail: Michelangelo seems to have always had them on, even when he slept. He only took them off when he changed them for a new pair, and his skin would even stick to the leather!