There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-geometric conjectures just to formulate the definition of motives.

There are things that I know about motives on some level, e.g. I know what the Grothendieck ring of varieties is or, roughy, what are the ingredients of the definition of motives.

But, how would you explain the Grothendieck's yoga of motives? What is it referring to?

So this is a crazy question, but I will try to give at least a partial answer. This question about the Beilinson regulator is also relevant, and this is also an attempt to reply to the comments of Ilya there. I apologize for simplifying and glossing over some details, see the references for the full story.

First of all, some references: A leisurely but still far from content-free exposition by Kahn on the yoga of motives is available here (in French). For Grothendieck's idea of pure motives, see Scholl: Classical motives, available on his webpage in zipped dvi format. For mixed motives, see this survey article of Levine. There is also lots of stuff in the Motives volumes, edited by Jannsen, Kleiman and Serre, here is the Google Books page. Finally, I would strongly recommend the book by André: Introduction aux motifs - this is has lots of background and "yoga", as well as precise statements about what is known and what one conjectures.

Pure motives

The standard way of explaining what motives are is to say that they form a "universal cohomology theory". To make this a bit more precise, let's start with pure motives. We fix a base field, and consider the category of smooth projective varieties, and various cohomology functors on this category. The precise notion of cohomology functor in this context is given by the axioms for a Weil cohomology theory, see this blog post of mine for more details. 

There are (at least) three key points to mention here: one is that a Weil cohomologies are "geometric" theories, as opposed to "absolute". For example, when considering etale cohomology, we are considering the functor given by base changing the variety to the absolute closure of the ground field, and then taking sheaf cohomology with respect to the constant sheaf Z/l for some prime l, in the etale topology. The "absolute" theory here would be the same, but without base changing in the beginning. In the classical literature, and in number theory, the geometric version is the most important, partly because it carries an action of the Galois group of the base field, and hence gives rise to Galois representations. On the other hand, the absolute version is important for example in the work of Rost and Voevodsky on the Bloch-Kato conjecture, and in comparison theorems with motivic cohomology. Similarly, it seems like cohomology theories in general come in geometric/absolute pairs.

The second key point to mention is that the Weil cohomology groups come with "extra structure", such as Galois action or Hodge structure. For example, l-adic cohomology takes values in the category of l-adic vector spaces with Galois action, and Betti cohomology takes values in a suitable category of Hodge structures. A nice reference for some of this is Deligne: Hodge I, in the ICM 1970 volume.

The third key point is that Weil cohomology theories are always "ordinary" in some sense, i.e. in some framework of oriented cohomology theories they would correspond to the additive formal group law (see Lurie: Survey on elliptic cohomology). If we allowed more general (oriented) cohomology theories, the universal cohomology would not be pure motives, but algebraic cobordism.

Now all these cohomology theories are functors on the category of smooth projective varieties, and the idea is that they should all factor through the category of pure motives, and that the category of pure motives should be universal with this property. We know how to construct the category of pure motives, but there is a choice involved, namely choosing an equivalence relation on algebraic cycles, see the article by Scholl above for more details. For many purposes, the most natural choice is rational equivalence, and the resulting notion of pure motives is usually called Chow motives. For a precise statement about the universal property of Chow motives, see André, page 36: roughly (omitting some details), any sensible monoidal contravariant functor on the category of smooth projective varieties, with values in a rigid tensor category, factors uniquely over the category of Chow motives.

Now to the point of realizations raised by Ilya in the question about regulators. Given a category of pure motives with a universal property as above, there must be functors from the category of motives to the category of (pure) Hodge structures, to the category of Q_l vector spaces with Galois action, etc, simply because of the universal property. These functors are called realization functors.

Mixed motives

It seems like all the cohomology functors one typically considers can be defined not only for smooth projective varieties, but also for more general varieties. The right notion of cohomology here seems to be axiomatized by some version of the Bloch-Ogus axioms. One could again hope for a category which has a similar universal property as above, but now with respect to all varieties. This category would be the category of mixed motives, and in the standard conjectural framework, one hopes that it should be an abelian category. It is not clear whether this category exists or not, but see Levine's survey above for a discussion of some attempts to construct it, by Nori and others. If we had such a category, a suitable universal property would imply that there are realization functors again, now to "mixed" categories, for example mixed Hodge structures. The realization functors would induce maps on Ext groups, and a suitable such map would be the Beilinson regulator, from some Ext groups in the category of mixed motives (i.e. motivic cohomology groups) to the some Ext groups which can be identified with Deligne-Beilinson cohomology.

We do not have the abelian category of mixed motives, but we have an excellent candidate for its derived category: this is Voevodsky's triangulated categories of motives. They are also presented very well in the survey of Levine. A really nice recent development is the work of Déglise and Cisinski, in which they construct these triangulated categories over very general base schemes (I think Voevodsky's original work was mainly focused on fields, at least he only proved nice properties over fields). 

To end by reconnecting to the Beilinson conjectures, there is extremely recent work of Jakob Scholbach (submitted PhD thesis, maybe on the arXiv soon) which seems to indicate that the Beilinson conjectures should really be formulated in the setting of the Déglise-Cisinski category of motives over Z, rather than the classical setting of motives over Q.

The yoga of motives involves far more than what I have mentioned so far, for example things related to periods and special values of L-functions, the standard conjectures, and the idea of motivic (and maybe even "cosmic") Galois groups, but all this could maybe be the topic for another question, some other day :-)
