First off, a few words on The Eternal Champion series: Each volume of the series tells the tale of a different incarnation of the 'Eternal Champion', in a different place and time, so you don't need to read them in order because they are different stories, and each time the champion is a totally different person. Although there a a few books that are sequels to previous volumes, 'Corum' isn't one of them. So, you can just jump straight in.
The Eternal Champion series takes place in Moorcock's multiverse that contains all the universes including our own. In this multiverse two forces have been battling for domination since the dawn of creation: The forces of order, and the forces of chaos. On each side gods, immortals, and puny mortals fight to gain victory for either order, or chaos. The 'Eternal Champion' is a soul that was destined (or cursed) to incarnate and fight for all eternity in all the universes in order to keep the balance between order and chaos. Since this is a balance issue, at times the 'Eternal Champion' will fight against order, and other times against chaos.
'Corum' unfolds in a universe where the forces of order just have lost a recent war and The Sword Rulers (a group of chaos gods) are taking over the world. The old races, the Vadhagh and the Nhadragh are becoming extinct because a new race (created by The Sword Rulers) is capturing their castles and defeating them with their greater numbers. The new race is basically the human race, however, the old races of this universe call them the Mabden. The main character is Prince Corum Jhaelen Irsei, one of the last Vadhagh. He is also known as The Prince in the Scarlet Robe, and he is also the present incarnation of the Eternal Champion, although he doesn't come to this realization until later in the novel. The story begins as Prince Corum sets off on a journey and returns to find his family's castle burned by Mabden and his family slaughtered. This incites Corum to begin quest that starts as nothing more than brutal revenge on the Mabden; especially the one who commanded the castle raiders, Earl Glandyth-a-Krae. Later Corum's revenge turns into a war to overthrow the forces of chaos, and to bring balance back to the universe.
Throughout his quest Corum learns many different concepts from his human enemies: hate, violence, cruelty, vengeance and slaughter; strange concepts to the old races but well known to human kind. He also falls in love with a Mabden woman, Rahlina. During his quest Corum will find the formidable magical weapons, The Eye of Rhynn and The Hand of Kwll and with them, he will travel the 15 planes of the universe (his universe), battling the Sworld Rulers; Arioch - The Knight of the Swords, Xiombarg - The Queen of the Swords and Mabelode - The King of the Swords , and also the lesser gods and their minions. Corum will meet many people on his way, some hostile and some friendly. Among them the immortal Jhary-a-Conel, Companion to heroes, and his winged cat, and also other incarnations of The Eternal Champion. Corum will even reach Tanelorn, the city that exists in all the universes, and within its borders the forces of chaos and order have truce.
Corum is a well written hero with a depth of personality that develops from a young naive Vadhagh prince, to an avenging lunatic, to a man who understands the nature of the multiverse who works to preserve balance between chaos and order. The other characters aren't as well written, and aren't really three dimensional (except for the great Jhary-a-Conel) but this isn't high fantasy, it's swords and sorcery (epic fantasy) and the focus is on the hero and not on the supporting cast. Beyond the constant fighting and bloodshed and meeting weird creatures, Moorcock debates: the nature of the universe; the relationship between gods and mortals; and human nature. Placing the humans, or Mabden as he calls them, as a nemesis who destroy everything in their path is an interesting choice that not many authors would attempt, and has send a clear message about his views of our world and society.