I don't know how to make this particularly comprehensive because the 12th SS Armored Division has a pretty short and sad history of underwhelming preformance. First things first, it was named Hitlerjugend because most of its original members were drawn from the Hitlerjugend born in 1926. These young men made up the enlisted ranks of the Division, the NCOs, senior officers and some combat experienced soldiers were drawn from other SS Divisions, primarily SS Panzer-Divison LSSAH.
The Divsion was sent to France in 1944 and took part in the counter-attack after D-day. It did not preform particularly well. Like many SS Divisions it was aggressive in combat but also suffered disproportionately high losses, something that had been a problem with the SS units ever since 1939. In a few short weeks of fighting in France the Division suffered nearly 80% casualties.
After its mauling in France the division was pulled off the line (with the exception of a single Kampfgruppe attached to SS Division Das Reich). It was reformed using mostly staff from the Luftwaffe and Kriegsmarine, a common practice at this stage of the war when reserves were in short supply. From this point on the Division was no longer mainly formed from Hitlerjugend members but retained it's name. It took part in the Battle of the Bulge/Wacht am Rhein where it again preformed underwhelmingly. It took part in the battle of Elsenborn Ridge where it failed to break through the American lines and lost the majority of its armor. During Wacht am Rhein the division again suffered over 50% casualties.
The division was again pulled from the front and reinforced with whatever was available for operation Spring Awakening. While it had some initial successes against the Soviets in Romania these successes were quickly reversed once the Soviets counter attacked in earnest. SS Division Hitlerjugend was pushed west ahead of the advancing Soviet armies and eventually reached Linz in Austria. It surrendered to the 65th US infantry division on may 8th of 1945.
The division is remembered primarily for its involvment in massacers of civilians and prisoners and for its heavy losses.