Dream World is where Pokemon can have abilities that they normally would not get otherwise. It's a very popular metagame on Pokemon Online. Let me list a few examples, based off of the usage statistics.

Chandelure gets Shadow Tag (probably going to be banned to Ubers in DW due to this, as it makes a number of Pokemon completely non-viable)

Tyranitar gets Unnerve - Preventing the opponent from using their berries can be useful.

Blaziken gets Speed Boost - Combined with Baton Pass, this has become one of the most used setup Pokemon.

Breloom gets Technician - Mach Punch being boosted. What more could you ask for? Not to mention the 30% paralyze Force Palm.

Conkeldurr gets Iron Fist - All punch moves being boosted. HURR DURR

You get the idea. These Pokemon can only have these abilities in Dream World.

Another thing worth noting that you may not know is that the Eviolite is an extremely useful held item that adds 50% to both Def and SpDef for ALL Pokemon that have an evolved form.

So that means it will work on Happiny and Chansey, but not Blissey. Chansey is actually used more than Blissey now, thanks to this.