Gohan is my favorite character other than Trunks. He is Goku's first son and a half-saiyan. Throughout the series of Dragonball Z, Gohan is shown to have hidden powers that exceed even that of his father's that are only awakened in times when the ones he loves are in danger. These powers are awakened in the following situations:
When Goku's evil brother, Raditz, begins to beat Goku up pretty bad, Gohan gets mad and escapes from Raditz's space ship and gives Raditz a headbutt in the stomach that cracked through his Saiyan armor. However, Gohan was not able to control his powers and does not succeed in destroying Raditz, rather just weakening him.
Later on in the series, Vegeta and Nappa arrive on Earth, also evil Saiyans. Gohan and his friends had been preparing for them but were still far to weak to defeat even Nappa who was much weaker than Vegeta. When Raditz is destroyed earlier in the series, Piccolo takes Gohan in to train under him. Nappa later throws a huge energy wave at Gohan and just as it is about to hit him Piccolo comes in and takes the blast for him. Gohan is enraged and does his own energy wave that was far stronger than he normally was, he was not however able to control his powers.
Next time is when Gohan is on planet Namek and Freiza kills his friend, Krillin. Though Freiza at the time was MUCH stronger than little Gohan, Gohan still launched an attack on Frieza that caught him off guard, it however did not destroy Freiza and Gohan was not able to control his power.
It happens again while at Namek after Goku arrives and transforms into the legendary Super-Saiyan, Goku being transformed is still weaker than Freiza and is nearly killed by Frieza. Upon finding out, Gohan goes to Freiza to stall him so the planet could be destroyed with Freiza still on it. Freiza discovers the plan and insults him, his dad, and the entire Saiyan race. Gohan then says "You hurt my dad, you killed my friend!" and again launches a huge attack even though Freiza was in his final form, it again did not kill him and again Gohan was not able to control his power.
The next time it happens is while training in the Room of Spirit and Time (Hyperbolic Time Chamber). Goku discovers these hidden powers in his son and knows that he will be the one to save the Earth from Cell (an evil android and the reason they are in there training in the first place). Goku, Vegeta, and Future Trunks have all become Super-Saiyans, Gohan desperately wants to help in the fight against Cell. Goku tells Gohan to use the pain of loss, he tells him to pretend that Cell is destroying everyone and everything, after a short while, Gohan says that he's never even seen Cell. So Goku says to think the same thing about Frieza. Gohan then is his own little world and is getting v. mad and succeeds in transforming into a Super-Saiyan.
After training with his dad for a while Gohan becomes as strong as Goku and after each are both v. tired and beat up Gohan says "No dad, lets train some more." and he powers up to a Super-Saiyan2 and then back to just regular Super-Saiyan then to normal form.
After Goku gives up against Cell, he appoints the next challenger, Gohan. Everyone can't believe Goku gave up and put his son in his stead. While Gohan fights Cell, he tells Cell to be careful and not push Gohan too far or else his hidden powers may awaken. Cell, being bored of no competition, wants to see these powers, so he makes these little Cell Jr.s to attack his friends. Just before Gohan's warning Android 16 tried to save Gohan, but ended up getting blown up by Cell; however, 16's head was still able to talk, and requests to be thrown to the battle field. Android 16 tells Gohan to protect the birds and plants that he loves so much, just then Cell says "Don't interfere." and smashes 16's head to pieces. Gohan completely loses it and his hidden powers are awakened and Gohan goes to Super-Saiyan2 and is finally able to control his powers and beats Cell up pretty bad, etc..
About 5 years after the fight with Cell, Gohan has devoted his time to his schooling and has not trained therefore getting weaker. While at the World Martial Arts Tournament he transforms to Super-Saiyan2 and Vegeta and Goku while still Super-Saiyan1 are stronger than Gohan due to his lack of training.
While trying to get away from Majin Buu he does get killed and does achieve the Mystic level while training with the Z sword when he's dead. Since the Mystic level is stronger than a Super-Saiyan 2, Gohan is always seen in his Mystic form while fighting rather than any form of Super-Saiyan he has achieved.
What forms Gohan as achieved and how:
Super-Saiyan: While training in the Hyperbolic Time Chamber, Goku tells Gohan to pretend Freiza has destroyed all his friends and the planet causing Gohan to remember the death of some friends at the hands of Freiza and also causing him to go SSJ1.
Super-Saiyan 2: While fighting Cell, Android 16 tells Gohan to protect the birds and plants he loves so much and gets his head crushed by Cell, triggering Gohan's anger to ascend him to SSJ2.