仇恨（Odium） is a Shard of Adonalsium, held by Rayse. The author of The Letter calls Odium the "most frightening and terrible of all the Shards." Odium had been on Roshar, though arrived on Roshar later than its other two Shards, Honor and Cultivation. Currently Odium is settled on Braize, one of the three Shardworlds in the solar system of Greater Roshar, but his presence is still felt on Roshar.
Over time, a person holding a Shard will be shaped to the Shard's intent. This can turn a kind person into someone who just wants to destroy. However, Rayse--Odium's holder--was purportedly already an incredibly crafty, loathsome, and dangerous individual, even before picking up the power. A Shard's intent is mildly interpreted by its holder (for example, the idea of Preservation could be interpreted in a number of ways, and the Shard's holder helps determine this), so Odium is probably being interpreted in the darkest, most dangerous way possible. Odium is a force of pure hatred which is held by a man who can--and will--use that power to the most terrible end.
Odium's ultimate goal is indeed terrible: he seeks to be the most powerful being in the cosmere. To this end, he kills other Shards (namely Aona, Skai, and Tanavast). But, instead of picking up those Shards (and becoming more and more powerful as he increases his Shard count), Rayse does not want to be unduly influenced from other Shard's intents. He feels that Odium matches him perfectly. So, he Splinters the Shards he kills, preventing anyone taking up those Shards to challenge him.
Odium intends to do this to every Shard and be the only Shard left, effectively making him unstoppable and giving him free rein in the entire cosmere. And, Odium appears to have no trouble with killing former acquaintances to progress its goals.
Perhaps unique to known Shards, Odium has figured out a method to kill a Shard's holder without destroying himself. Compare this to Preservation, which in order to defeat Ruin, had to kill herself to do so. While that may be particular to Ruin and Preservation's circumstance of being perfect opposites, there is no evidence of Shard death aside from Ruin and Preservation, and Odium's power. While how Odium accomplishes this is a mystery, It is known to be very taxing and take a long time to recover from.
The first theory to explain this was that Odium was inherently more powerful as a Shard, and so he can simply overpower other Shards. This is not the case, however, since all Shards initially had the same power level. It is possible that since Odium is unlikely to invest his power in granting to sentience (as Preservation did), Odium is now slightly more powerful than his opponents, but this still does not explain how Odium easily could defeat Aona and Skai's Shards, Devotion and Dominion.
Another explanation could be that potentially any Shard could figure out Odium's destructive methods, but lack the motivation compared to Odium's overpowering intent of hatred.
Odium's other known ability is his power to Splinter Shards. This appears to require the opponent's Shardholders to already be dead before Odium can Splinter them. It is unclear if any Shard can Splinter Shards, or it is especially unique to Odium.
Odium's actions are more obvious on Roshar, his present location. Honor stated to Dalinar that Odium is the enemy, and implied he is the cause of the Desolations and potentially the leader of the Voidbringers. So, Odium has been attacking societies on Roshar for over 4,500 years. This ultimately culminated with Odium killing Tanavast and Splintering Honor, though the holder of the third Shard on Roshar, Cultivation, still lives. In the present, Odium brings about the threat of the Everstorm and the True Desolation.
Odium may be the cause of the Heralds being tormented after dying in the Desolations.
Odium has killed at least one other unknown Shard.
Odium may be the one referred to as the Broken One. This quote mirrors how the voice in the storm talked to Kaladin and said "Odium reigns". The "sixteen" surely indicates the sixteen Shards, and indeed there are three Shards on Roshar. However, it is peculiar that if three reigned, why is Odium (presumably one of the three) referred to so strangely as the "Broken One" here? It is possible to say that indeed Odium is one of the three mentioned rulers and as to why he is called the broken one it can be said that while Odium ruled along with the other two shards he was still solid and "whole" in the sense that he was of physical form, But as indicated by Wit he is more of a force than a living being so this Broken one status we can say is him becoming this force
It has been theorized that Odium is much better at seeing into the future than Ruin was. It is postulated that a Shard's skill in seeing the future is directly related to their intent and the mindset it gives them. Under this theory, the reason Ruin was very bad at seeing the future is that he always desired to destroy immediately, and so, he was focused on the present much more than Preservation, who had to think much more long term to fulfill his intent.
Odium has shown much craftiness and planning, so he probably is significantly more adept at seeing the future than Ruin. It isn't clear in this theory just how good Odium is at future sight--it is entirely possible that Preservation and especially Cultivation are still superior to Odium in this regard. Still, if Odium is better at seeing the future than Ruin, that already makes Odium a much bigger threat than Ruin ever was.
During battles, Dalinar sometimes feels disgust about fighting. This may be in relation with this.