What's the relation between Mansions of Madness, Elder Sign and Arkham Horror?
Mansions of Madness, Elder Sign, and Arkham Horror are all games whose themes pull from the Cthulhu Mythos.

In Elder Sign and Arkham Horror you play as investigators cooperating in an attmpt to prevent/drive away an Ancient One entering this world, but the mechanics of both games are very different. I'll describe the games generally.

Arkham Horror has you play investigators wandering around the town of Arkham, having encounter fighting monsters, traveling through gates to other worlds, and sealing said gates. When you seal enough gates (or some other less often-fufilled victory condition), you win. Game play is driven by skill checks. Each investigator has 6 skills and will use them (along with potential skill boosting items) to make skill checks to succeed at various tasks (Understand lore, escape a house before it collapses, kill a monster, etc.)

Similarly, Elder sign has you playing these very same investigators trying to prevent the ancient horror from entering the world, but there's no board to speak of. Instead, your turn consists of picking one of the encounter cards on the board, gathering a pool of specialized dice, and rolling them, trying to match the necessary results displayed on the cards. Winning gets you various rewards. You win when you get enough 'Elder Sign' tokens, which could be direct rewards from completing a mission or by cashing in some of the trophies you've earned.

Mansions of Madness pits a team of investigators against a 'Keeper', who is trying to fufill some hidden agenda of his. The investigators explore a pseudo-randomly generated building trying to discover, and then prevent, the Keeper's goal. The Keeper meanwhile tries to nudge the investigators in the right direction so he can fufill his secret goal easier. The 'board' in this game consists of modular parts that get rearranged depending on the particular scenario being played.

So mechanically and piece-wise, the games are very different. The only pieces you can use in any of the other games are the markers for the investigators. 

In Arkham Horror, the investigators have standing cardboard-y pieces with their images on them. In Elder Sign, the investigators have smaller cardboard-y pieces with their images on them. In Mansions of Madness, the investigators have moulded plastic figures in their likeness. Any of these markers could be used in any of the games equally as effectivelly.

Additionally, the monster figures from Mansions of Madness can be used in Arkham Horror. Arkham Horror's monster tokens are squard cardboard pieces with various monster fighting statistics on them. In Mansions of Madness you have the same sized tokens, but the printed stats are totally different, making them non-interchangable. But Mansions of Madness has you slot these tokens into larger monster figures (the tokens are still readable if you look under the base). So you can use these figures for your Arkham Horror play.

Finally, the figures that come with Mansions of Madness are all unpainted. Fantasy Flight sells painted versions of all of these figures, and more, on their website. So if you like the idea of having investigator minis, but don't want Mansions of Madess, you can still get whichever figures you like.