Altmer looks down on other "inferior"/cursed/half-breed mers. Various mers share different religions which can bring up some conflicts.
Well, Ondolemar referred to the Bosmer as "brethren" and "cousins".
As for the Dunmer, it's always a safe bet that they hate everyone.
As I understand it, only the Altmer and Bosmer share some mutual respect or affection on a general level. Dunmer pretty much hate everyone else, with particular emphasis on beast races. Orcs are forever outcasts, and are understandably reclusive from, and distrustful of, outsiders.
Racial and Faction reaction tables. Oh how I miss those...
I would say Altmer hates any non-Altmer that breathes.
You may not be reading the table correctly. The table indicates that if you are playing an Altmer, Dumner or Orc, there is a negative reaction from Bosmers.
The Altmer and Dunmer have had a rivalry that goes back since before the Dunmer were Dunmer. The prophet Veloth led a group of Aldmer out from their homeland and into present-day Morrowind, where they came under the influence of the so-called "good daedra." That was pretty much it between them and the Altmer. The god Trinimac tried to bring them back to the fold, so Boethiah ate him and spit him out, changing his followers into orcs. That was pretty much it between orcs and Dunmer, too, and orcs and Altmer. Meanwhile the Chimer (as the Dunmer were called then) also fought with the Dwemer. Other Mer considered Dwemer heretics because they renounced all religion as the Mer knew it, prizing reason and logic instead.
There's a good story called http://www.uesp.net/wiki/Lore:Chimarvamidium about the Chimer-Dwemer wars. The author's a fraud but as the note at the end mentions, this one likely does have elements of truth.
And, as someone mentioned, the Maormer (sea elves from the island of Pyandonea) and Altmer have fought back and forth for ages. The Ayleids also fought amongst themselves, which is probably one reason Alessia's rebellion succeeded. The Altmer also have had civil conflicts and naturally the Dunmer Great Houses were always squabbling and killing each other off.
Us wood elves think all others have lost their way.
(Michi's opinion, not lore)
the forests are hard to navigate through, so let's burn it all down.
Us orcs have a well established way of life. All we want is that other species (being them pure, or not, superior or not), to just leave us alone. We don't want to be involved in complicated political struggles, we are not the The Orchis Horde from other fantasy RPGs, we do not want to conquer, enslave or plunder everyone in Tamriel. We just want to live our life as we learn from our ancestors, following The Code.
The schism between Altmer and Dunmer is ancient history. I doubt they hate each other, but I think they'd both consider the other pathetic degenerates. And Bosmer would fit into that category too, but they'd be also be called backward which gives them an excuse. They looked more redeemable for other reasons, mostly to do with their Aldmeri culture, spirituality and less "effete" way of life. There were Orcs leaving peacefully enough in Valenwood before their race was accepted across the rest of Tamriel, but then they had centaurs and Imga too.
There's dialogue in Morrowind in which their hatred of different races is actually ranked, but only their neighbours. It went Nords, Imperials and Argonians in ascending order. I remember a slave trader calling Bosmer the least of the Elven races (that probably didn't include Orcs).
Altmer: Dislike Orsimer and Dunmer due to their relations with Daedra somewhat neutral towards the bosmer (Besides the Thalmor) but have warred with them often in the past and absolutely hate the Maromer.
Dunmer: Dislike Orcs and Altmer and I would guess pretty neutral towards Bosmer but have fought them in the past.
Orsimer: Well the others don't like them much and they don't like them in general but don't really hate any of the other elves and more then they do anyone else.
Bosmer: Eh they usually seem neutral with the Altmer most of the time but have fought wars against them often in the past. (Besides the Thalmor) pretty neutral towards dunmer and orcs in general due to lack of interaction outside of valenwood I would assume.
Maromer: Hate everyone and view them as inferior especially the Altmer. (You'll see lot of them if you play dominion side in ESO)
Falmer: Well the Dwemer have made sure that they despise other elves as much as they do men even as we see in dawnguard their uncorrupted kin.
Khajiit: Well they seem to be rather neutral to every one but Bosmer and Dunmer. Being a beast race generally looked down upon by all other Elves and a lot of men.
I seem to recall some dialogue in skyrim by a thalmor guy which suggested bosmer can actually be in the thalmor?
That markarth guy says something like "the thalmor are agents of the aldmeri dominion which is an alliance of altmer and our cousins the bosmer" or something like that?
Apart from that one line they seem 100% altmer and like they'd hate everyone else, but yeah he says something like that.