Note: I am not sure that each Primarch has a defined role after the Great Crusade and the complete unification of the Galaxy as the Imperium of Mankind. Besides being a charismatic leader and a deputy for the Emperor (He cannot be everywhere!) of course.
Some had a definite purpose, but it is hard to tell if it was planned by the Emperor from the start, an inevitable consequence of their creation that each of them would be heavily specialized, a consequence of their dispersion by the forces of Chaos and/or their upbringing on various worlds. Fittingly, the powerful psyker Magnus grew up on Prospero, a world far from Terra where psykers could hide and create a society emphasizing knowledge, while Gulliman was educated on a planet where the elite were keen on discussing politics and law.
More than having a specific purpose designed by the Emperor, each Primarchs represents an aspect of the perfect soldier and ruler the Emperor is. We should remember that 40k is before anything a game and, from a gameplay point of view, the philosophy of Games Workshop is to create a generic, vanilla version of each race and army, and give players enough flexibility to play with their own style. This is embodied later in the fluff. The Eldars Craftworlds, the Orks clans, the Imperial Army regiments are examples of this philosophy. The purpose of the historical articles J-files by Jervis Johnson was to encourage players to free their creativity and create the fluff for their own army.
That said, we can try to detail what aspects/purpose each Primarch embodies.
Note 2: that would be a lot to link to references for anything in that answer. The general references are Lexicanum and Warhammer Wikia. I would also have difficulties to quote the books, as I read them in French.
- Horus: obviously, military leadership and charisma. There were tensions between some Primarchs from the early days of the Great Crusade, but all of them liked Horus and were close to him to some extent.
- Magnus: the most powerful pskyer after the Emperor. Magnus was destined to seat on the Golden Throne and control a portal to the Eldar Webway, freeing mankind from having to use the Warp to travel through the Galaxy. He is also a personification of erudition and knowledge.
- Fulgrim: perfectionism and quest for perfection. Fulgrim and his Emperor's Children, especially Lucius, were constantly trying to improve their abilities as soldiers.
- Perturabo is the master of siege. Also, it is explained in Perturabo (in the Primarchs series) that one of the reasons that the Iron Warriors campaigns and sustain more losses than the other Legions is that the Emperor choose them for the most difficult missions. Perturabo then embodies tenacity.
- On the contrary, Rogal Dorn is the master of fortifications and defense. This includes planning before a battle. In addition, Rogal is known to be stern and to embody rigor.
- Roboute Gulliman also embodies planning and organization, but more as a leader and a ruler. It is said (I think in The First Heretic) that no world conquered by the Ultramarines during the Great Crusade ever revolted against the Imperium.
- The First Heretic explains that Lorgar is the Primarch that physically resemble the most to the Emperor. Lorgar represents faith and devotion to a cause (at first to the Emperor, later to the Gods of Chaos).
- As said in the OP, Leman Russ is the Emperor's executioner. In addition of being tasked of punishing the Thousand Sons, it is (not that) subtly hinted that the Space Wolves were tasked to eliminate the Second and Eleventh Legions. As a warrior, Leman Russ represents the rage of the warrior.
- Angron: difficult to tell as Betrayer explains that Angron was broken by his upbringing as a gladiator and the Butcher's Nails that constantly excite his nerves. A scene that I found great in Galaxy in Flames is when Lunar Wolves and World Eaters meet and Garviel Loken (I think) realize how much willpower is necessary for the World Eater to control their inner rage. To me, Angron initially represented willpower and control of emotions (in this scene, this aspect is directly put in opposition with Space Wolves' hot temper).
- Konrad Curze is another unknown. Not clear if he had the power of precognition from his creation, or if he was tainted by the forces of Chaos when the Primarchs were dispersed across the galaxy. In the first case, we may hope that being raised by the Emperor would have prevented Curze from being insane and the power of precognition would have proved handy.
- Vulkan was the master of forges. He represents the knowledge of weapons. Also, Vulkan Lives explains that, among all Primarchs, Vulkan is the one that has the truest and deepest love for his children and mankind in general. Vulkan is the symbol of love and devotion.
- [Edited] Ferrus Manus has a lot in common with Vulkan, in the sense that he is an expert in technology. But contrary to Vulkan, he abhors weakness. After Istvaan disaster, this leads his Iron Hands to understand the Legion's motto "The flesh is weak" too literally and use bionics more than any other Legion, while Ferrus Manus implied a search for improvement by understand human nature and body (thanks to TommyB for the link). Ferrus Manus represents also perfectionism. The fact that he waited to address the bionic problems with his Legion seems to imply a distance and difficulty to relate to inferior people, similar to the Emperor (how come the Emperor never told his master plan to anyone before everything was spoiled!)
- Not much is said so far about the stern Mortarion in the Horus Heresy series. He represents resilience and endurance.
- Corax is the master of stealth and blitzkrieg.
- Alpharius/Omegon represent sabotage and intelligence.
- More than any other Primarch, Sanguinius represents perfection and charisma. He encouraged Blood Angels to study arts and, as the Emperor, is an idealization of mankind.
- Lion El'Jonson: hard to tell. Lion El'Jonson has a lot in common with Horus, the main difference being that because of his cryptic behavior of the Lion (see Fallen Angels) and his arrogance, the Lion is seen as distant and didn't gain as much trust as Horus.
- To me, Jaghatai Khan and Leman Russ have very similar personalities. The main difference is that Leman Russ had no problem with being the Emperor's executioner while I think the Khan would have reluctantly accepted such a task. Also, as Claws explains, Jaghatai Khan values freedom and independence more than anything, which lead him to have difficulties to connect to other Primarchs and difficulties to choose a side at the beginning of Horus Heresy.
- All information about the Primarchs of the Second and Eleventh Legions have been expurged from archives, and thus nothing is known about them.
Finally, it is difficult to assign a specific role for each Primarch, as the authors tend to oppose two Primarchs, one loyalist and one traitor, by giving them either opposite traits, or similar traits. For example, Blood Angels and World Eaters are both specialists of direct assaults.