If you look at the Dr. House video, all of the figures look to be 2D drawings, but they appear to be moving in a 3D space - many even giving the illusion of turning in three dimensions. This is accomplished through clever layering, shadows, perspective animations, morphing, and it looks like some of the animation is done using bone systems. This creates a certain fluidity and changes in depth that make us think we're looking at 2D paper cutouts moving in a 3D space, almost like paper dolls standing on a table.
The 2.5D effect also extends to objects that don't look like paper cutouts, but just typical 2D animations - but they appear to move in a 3D space. This can be as simple as a character's head turning, but it moves with that 3D fluidity that makes it seem like a 3D object with flat shading and outlines to give it a 2D look, and further creates the impression of 3D space around a 2D object. You see this a lot in newer episodes of Family Guy, where they actually do use 3D animations rendered to look like 2D animations. They're 3D objects moving in a 2D space.
Adding shadows to a 2D object can also accomplish the 2.5D effect. I don't mean shading the figure itself; I mean having it cast a shadow on the objects around it. Think of a 2D object standing against a white background. On its own, it's just a drawing on paper. Flat. Two-dimensional. Now add a shadow of that object, stretching into the distance - into the paper, a dimension that technically doesn't exist for this object but that we can imply. Suddenly it's a 2D object standing in a 3D space, because we've created the implication of a third dimension by adding this shadow - which should taper off with perspective to further imply depth.
I've seen people create 2.5D effects in both 2D and 3D programs; in programs such as Flash, you can use the tools to create false perspective, and then animate / tween that perspective changing for a 2.5D effect. You can also create shadows using copies of objects greyed out and arranged properly, with perspective tools appropriately stretching the shape. In 3D programs, you can map 2D objects onto transparent polygons and then animate and position them, with lighting to add shadows.
No matter which way it's done, if you look closely, you'll see 2.5D everywhere.