Each race probably has its own language, certainly, we've heard words in several languages. However, it seems likely that unless said otherwise, all dialog in the game is rendered in Cyrodiilic, or whatever other language is most commonly used in the location the game takes place in. After all, no one ever has trouble understanding anyone else unless the plot requires it, and it would be pretty impractical if everyone had to learn ten different languages just to be able to comprehend what everyone they meet says. Also, considering that the Empire controls all of Tamriel, at least at the time all the games in the main series have so far taken place, it would also seem to make sense that its language would spread to all these areas, even if not everyone is necessarily fluent in it (Possibly this is the source of Khajiit speaking in third person? Maybe their language just doesn't use first person pronouns, that's my theory, at least.)
As in so many other cases, it should be stressed that the games are obviously only a translation of the actual fictional word, not a perfect copy. Any statements that ignore that very basic fact are inherently flawed.
While that is true, if Bethesda were feeling ambitious, they could have had the voice actors speak the language of their race and than have subtitles in English, or whatever language the version your playing is in (At least in Oblivion, in earlier games, this would encounter the problems of most NPC lines not being voiced. However, there are ways one could reflect that dialog is spoken in a language other than whatever audiences are seeing it in even when it's only written.) had they wanted to reflect that the characters were actually using their own language, in general, in fiction if you're not told someone is using a different language, you can assume that they use whatever language everyone else uses. Just as fiction is liable to render dialog on foreign or fictional languages in plain English (or whatever language the work was made in or translated into.) so that audiences can easily understand it, it may also put in cues to tell the audience that this is the case, due to http://tvtropes.org/pmwiki/pmwiki.php/Main/RuleOfPerception, which would dictate that if you want the audience to know something, you should show them that this is the case. In other words, if characters are speaking their native tongue, then the game would show us this fact somehow.
It should also be pointed out that it seems hard to believe that most dialog is rendered in the native language of the characters' race. Considering that the player character can always understand their dialog, does the player character know all languages commonly used in Tamriel? I might be able to believe that if the player character were a learned scholar, but what if you're playing as a barbarian? Obviously, from a gameplay standpoint, the game needs to render dialog in a language the player can understand so you don't need to learn a whole fictional language for this purpose, however, since the player character is like the player's avatar in the Elder Scrolls world, if the player is meant to be able to understrand something it can be assumed that the player character can too, if the player character cannot understand something, it would not be rendered in a language the player is expected to understand. And if NPCs ever had any trouble understanding each other, this would be noted in the game, if there are no actual NPC conversations where one party can't understand the other, than perhaps characters who are not native to the location the game takes place in would complain about being unable to understand the locals, this never happens, therefore this suggests that everyone can understand each other without problems.