The poem, which in detail bears almost no resemblance to the actual history or cultural setting of the Crusades, tells of the initial disunity and setbacks of the Christians and their ultimate success in taking Jerusalem in 1099. The main historical leaders of the First Crusade feature, but much of the poem is concerned with romantic sub-plots involving entirely fictional characters, except for Tancredi, who is identified with the historical Tancred, Prince of Galilee. The three main female characters begin as Muslims, have romantic entanglements with Christian knights, and are eventually converted to Christianity. They are all women of action: two of them fight in battles, and the third is a sorceress. There are many magical elements, and the Saracen side often act as though they were classical pagans. The most famous episodes, and those most often dramatised and painted, include the following:
Sofronia (in English: Sophronia), a Christian maiden of Jerusalem, accuses herself of a crime in order to avert a general massacre of the Christians by the Muslim king. In an attempt to save her, her lover Olindo accuses himself in turn, and each lover pleads with the authorities in order to save the other. However it is the arrival and intervention of the warrior-maiden Clorinda which saves them (Canto 2).
Clorinda joins the Muslims, but the Christian knight Tancredi (in English: Tancred) falls in love with her (Canto 3). During a night battle in which she sets the Christian siege tower on fire, she is mistakenly killed by Tancredi, but she converts to Christianity before dying (Canto 12). The character of Clorinda is inspired in part by Virgil's Camilla and by Bradamante in Ariosto; the circumstances of her birth (a Caucasian girl born to African parents) are modeled on the lead character (Chariclea) from the ancient Greek novel by Heliodorus of Emesa.
To prevent the crusaders from cutting timber for siege engines, the Muslim sorcerer Ismen protects the forest with enchantments, which defeat the Christian knights, even Tancredi (Canto 13). Eventually the enchantments are broken by Rinaldo, and the siege engines built (Canto 18).
Another maiden of the region, the Princess Erminia (or "Hermine") of Antioch, also falls in love with Tancredi and betrays her people to help him, but she grows jealous when she learns that Tancredi loves Clorinda. One night she steals Clorinda's armor and leaves the city, in an attempt to find Tancredi, but she is attacked by Christian soldiers (who mistake her for Clorinda) and she flees into the forest, where she is cared for by a family of shepherds, with an old man who weaves baskets (Cantos 6-7).
Later in the poem we find her again in the company of Armida's ladies, but Erminia abandons her Muslim people and goes over to the Christian side. When Tancredi is dangerously wounded in combat, she heals him, cutting off her hair to bind his wounds (Canto 19).
The witch Armida (modeled on Circe in Homer and the witch Alcina in Ariosto's epic) enters the Christian camp asking for their aid; her seductions divide the knights against each other and a group leaves with her, only to be transformed into animals by her magic (Canto 5).
Armida comes across the sleeping Rinaldo, the greatest of the Christian knights, and abducts him in her chariot (Canto 14). He has the same name as a Carolingian paladin count who is a character in Ariosto's Orlando Furioso [III, 30]; he is the son of Bertoldo and was the reputed founder of the House of Este. She intends to kill him but she falls in love with him instead and takes him away to a magical island where he becomes infatuated with her and forgets the crusade. Carlo and Ubaldo, two Christian knights and close companions of Rinaldo, seek out the hidden fortress, brave the dangers that guard it and find Rinaldo and Armida in each other's arms. By giving Rinaldo a mirror of diamond, they force him to see himself in his effeminate and amorous state and to return to the war, leaving Armida heartbroken (Cantos 14-16).
Rinaldo is deposited on a shore where he finds a shield and sword, and the "Mago d'Ascalona" ("Wizard of Ascalon") shows him a vision of the future in the shield, including the glories of the House of Este (Tasso drops in several prophecies of the time between 1099 and his own at various points). Rinaldo resolves to pursue the crusade with all his might (Canto 17).
Armida is grief-stricken and raises an army to kill Rinaldo and fight the Christians, but her champions are all defeated. She attempts suicide, but Rinaldo finds her in time and prevents her. Rinaldo then begs her to convert to Christianity, and Armida, her heart softened, consents (Canto 20). (This sequence echoes a similar storyline in Ariosto: the witch Alcina ensnares the knight Ruggiero, but the spell is broken by a magic ring that the good sorceress Melissa brings him; earlier antecedents include Calypso's attempt to keep Odysseus on her island Ogygia and Morgan le Fay taking Ogier the Dane off to a faraway island.)
After the enchantments on the forest are broken, finally the Crusaders breach the walls and take the city, with some Muslims remaining in the Temple Mount. But an Egyptian army is known to be arriving in a few days (Canto 18). When they arrive there is a great battle outside the walls, which the Christians win, completing their quest (Canto 20).

Question: Where are the Christians trying to take over?
Answer: Jerusalem