Khajiit:
Khajiits may not be "hated," but are "disliked" by most races because of their reputation as thieves and smugglers.
Khajiits (I think) were the founders of turning moon sugar into Skooma, which is basically the opiates of TES. (Look in Miaq The Liar's pockets if you want to laugh)
Honestly, I can't think of a race that really enjoys having Khajiits around, but the Argonians at least relate to them in matters of appearance, because let's face it, they are both pretty ugly compared to the other races. (Not that appearances has anything to do with lore.)
Argonians:
I've heard Argonians were enslaved by the Dark Elves, which obviously means there are some bad blood there, and I know Nords of Skyrim (Especially Stormcloaks) don't much care for them.
I really don't know much about them except for their sacred tree (Don't remember the name) that made them excellent warriors.
Bretons:
Breton is my favorite race to play in any ES game. However, I unfortunately have not done enough research to know who hates them. If I were to make an educated guess, I would say because of their ability to use magic (not as well as the Altmer) would cause the High Elves to look down at them.
High Elves:
Again, with the bad blood between the Dark Elves and the High Elves, because the High Elves enslaved them, they hate each other, and I don't think that's going to change.
Also, the High Elves got into a war which I think is called the "Great War" with the Nords, which explains why Nords dislike magic users. The war itself caused the Nords to despise the High Elves.
Nords:
The Nords aren't hated by anybody that I can think of (Besides in Skyrim for going against the Imperials and the High Elf Thalmor Embassy), but they are usually tolerated as drunken brutes by the other races. I don't really know much else about them.
Redguard:
Yeah, I don't know squat about them, where they live, who they hate or who hates them.
Wood Elves:
Eh. I know these guys lived in trees for many years and are able to befriend any animal that approaches them, but as far as who they hate or who hates them, again I'm stumped.
Dark Elves:
I've already went over the HE/DE lore enough, I'm not going to repeat it again.
As far as hating anybody else besides Altmer or Argonians, my knowledge doesn't extend that far.
Imperials:
In Skyrim, this is obvious that the Nords are upset with the Imperials for trying to take over.
I think they were in a war with another race, but I don't know which one or who it was against.
Orcs:
I don't know much about these guys either. I know they were in a war, but again I don't know who it was against.
I know orcs usually keep to their own race, and you usually have to prove yourself before they let you in their circle of trust, but I don't know why they do that.
As you can see, I've got a lot to learn about this stuff. Hopefully one you can shed the light on it for me.