In Luke and Matthew's accounts, the devil tempts Jesus to:

1) Worship the devil in return for all the kingdoms of the world. Luke has the devil explicitly claim this authority had previously been handed to himself, the devil;

2) Make bread out of stone(s) to relieve his own hunger (and He was really hungry);

3) Free himself from a pinnacle by jumping and relying on angels to break his fall. The narrative of both Luke and Matthew has the devil quote Psalm 91:11-12 to show that God had promised this assistance, although the devil omits the part of that passage which makes clear that it is only accidents that are being referred to, not deliberate jumps.

Why did Christ allow Himself to be tested??
What can we learn from Jesus' outright rejection of these (mysterious) temptations

Thomas Aquinas gave four reasons why Christ allowed Himself to be tempted:

1) First that He might strengthen us against temptations. Hence Gregory says in a homily (xvi in Evang.): "It was not unworthy of our Redeemer to wish to be tempted, who came also to be slain; in order that by His temptations He might conquer our temptations, just as by His death He overcame our death."

2) Secondly, that we might be warned, so that none, however holy, may think himself safe or free from temptation. [/b]Wherefore also He wished to be tempted after His baptism, because, as Hilary says (Super Matth., cap. iii.): "The temptations of the devil assail those principally who are sanctified, for he desires, above all, to overcome the holy. Hence also it is written (Ecclus. 2): Son, when thou comest to the service of God, stand in justice and in fear, and prepare thy soul for temptation."

3) [b]Thirdly, in order to give us an example: to teach us, to wit, how to overcome the temptations of the devil. Hence Augustine says (De Trin. iv) that Christ "allowed Himself to be tempted" by the devil, "that He might be our Mediator in overcoming temptations, not only by helping us, but also by giving us an example."

4) Fourthly, in order to fill us with confidence in His mercy. Hence it is written (Heb. 4:15): "We have not a high-priest, who cannot have compassion on our infirmities, but one tempted in all things like as we are, without sin."