The Lion received the Emperor's honor, which made him level-headed enough to be a tactical genius as well as diplomatic and sincere enough to lead men effectively. Unfortunately, this also gave him a personality flaw which made him make questionable decisions on occasion.
Fulgrim received the Emperor's desire, which made him seek perfection in all things and gave him the capacity to appreciate the beauty in things others would consider hideous. Unfortunately, wanting to push his own boundaries made him susceptible to doing questionable things in order to do so.
Perturabo received the Emperor's creativity, which made him seek democracy and diplomacy as alternatives to rule of force and warfare, and made him a genius architect. Unfortunately, daddy didn't want a diplomat, and forcing him to be a siege engine gave him an inferiority complex to Dorn, as nothing he was forced to build could ever be as good as something Dorn was made to build.
The Khan received the Emperor's wanderlust, which made him an ideal crusader as he always wanted to see what was beyond the next horizon. Unfortunately, he didn't care too terribly much about his brothers and their problems, as those things only served to keep him from moving forward.
Leman Russ received the Emperor's instincts, which made him handy in a tight spot and able to see to the heart of a matter. Unfortunately, this left him relying less on rational thought, which resulted in his charging into places without contemplating why.
Rogal Dorn received the Emperor's fortitude, which made him psychologically stable enough to withstand just about everything that could be thrown at him. Unfortunately, this stability made him utterly inable to use tact, as his mind just wasn't flexible enough in social situations to realize its benefit.
Konrad Kurze received the Emperor's insight, which made him understand human psychology and gave him his visions of the future. Unfortunately, understanding the psychology of humans combined with his ability to predict the future made him something of a neurotic mess who sought to impose order on the chaos through whatever means were necessary.
Sanguinius received the Emperor's compassion, which made him probably the most likeable Primarch and gave him an appreciation for the humans he was supposed to be championing. Unfortunately, this left him unable to destroy a once dearly beloved foe, and led directly to his fall.
Ferrus Manus received the Emperor's diligence, which made him extremely hard-working and efficient at the tasks he set his hands to. Unfortunately, this made him distant from all but very few of his brothers, and prevented him from curtailing the cybernetic excesses of his Legion.
Angron received the Emperor's courage, which made him able to stand against insurmountable odds when he knew the cause was right. Unfortunately, that same courage turned into recklessness when combined with the Butcher's Nails, leading him down a path of unrelenting fury and bloodshed.
Guilliman received the Emperor's patience, which made him able to think long-term and create vast strategies which are still in use in the 41st millenium. Unfortunately, he came off as egotistical to those of lesser patience (that is, everyone else) and his tendency for detailed planning left him holding the bag when unexpected events occurred.
Mortarion received the Emperor's self-reliance, which made him the toughest, strongest SOB out of all of the Primarchs as a rule. Unfortunately, he considered anyone not strong enough to make do on their own less than human, and he extended this thought to anyone who used the Warp to accomplish things, making him view the Emperor as a hypocrite.
Magnus received the Emperor's curiousity, which made him eager to study and learn all that he could about everything. Unfortunately, this is Warhammer 40K.
Horus received the Emperor's ambition, which made him the ideal Warmaster to take the Emperor's dreams to the stars. Unfortunately, one cannot have ambition without a certain amount of vanity, without believing that the things you're striving for are, in a sense, owed to you. Horus loved the adoration he received for being Warmaster, and couldn't conceive of his father's vision of an Imperium which didn't need him or his brothers as leaders anymore, leading him to oppose the Emperor in an effort to claim all the glory all the time. Much is made of how seemingly easy it was for Horus to be convinced to side with Chaos, but in truth all the Ruinous Powers did was provide him with backup and "justification" for what he really wanted to do anyway.
Lorgar received the Emperor's faith, which, as the Emperor's weakest and most repressed personality trait, made him the physically weakest Primarch, but the one who had the most potential for empathy. Unfortunately, the Emperor wanted nothing to do with faith and refused to acknowledge that humanity needed some level of belief in a higher order, thus setting up Lorgar to be the instrument of his downfall.
Vulkan received the Emperor's humanity, which made him the Primarch most concerned with the little people and an absolute beast when it came to defending them. Unfortunately, he didn't really believe that he was actually worthy of commanding people, and thus while a good commander, he was too kind-hearted for war.
Corax received the Emperor's romanticism, which made him into the quintessential Underdog who used guerilla tactics and small, elite forces to bring justice to larger, stronger forces of evil. Unfortunately, this is 40K.
Alpharius Omegon received the Emperor's mind, which made them scheming bastards par excellence, but also made them think that they were right all the time and that they were better than everybody else.