In one sentence: the theory of cohomology theories on algebraic varieties and the idea that there is a universal such thing.
Of course, this is not a very satisfying answer, unless we specify what a cohomology theory is. Examples are l-adic cohomologies, singular cohomology, de Rham cohomology, Deligne cohomology, rigid (or Monsky-Washnitzer) cohomology. The idea is that any computation which seems to hold in all these nice cohomology theories should be motivic (which means that it should be obtained from the analogous computation in the (conjectural) category of motives by the suitable realization functor): example of such computations are those which involve only intersection theory (using cup products of cycle classes). Conjecturally, the theory of motives is essentially determined by intersection theory of schemes, while higher Chow groups (i.e. motivic cohomology) should be to motives what Deligne cohomology is to mixed Hodge structures.
1) Pure motives --- Historically, such a cohomology theory was thought as one which behaves like singular cohomology (with rational coefficients) or de Rham cohomology on smooth and projective varieties over complex numbers, so that we would have cycle classes, the Künneth formula, Gysin maps as well as the projective bundle formula (hence Poincaré duality), and, as a consequence, a Lefschetz fixed point formula (i.e. everything needed to do intersection theory there). If the cohomology theory takes its values in the category of (graded) vector spaces over some field of characteristic zero, this leaded to the notion of Weil cohomology (they were named after Weil because of his insights that the existence of such a cohomology for smooth and projective varieties over a field of characteristic p>0 would imply the Weil conjectures, i.e. the good behaviour of the zeta functions associated to the Frobenius action). However, there is not any universal Weil cohomology: over finite fields, the existence of l-adic cohomologies would imply that this universal theory would be with coefficients in the category of Q-vector spaces, and it is known that that there is no Q-linear Weil cohomology for varieties over a field k which contains a non trivial extension of the field with p elements (this follows from a computation of Serre which shows that supersingular elliptic curves over such a k cannot be realized with Q-linear coefficients). (One of) the observations of Grothendieck was that, in practice, Weil cohomology theories takes their value in more complex categories, namely tannakian categories (e.g. Galois representations, mixed Hodge structures), which made him conjecture the existence of a universal cohomology theory with values in a tannakian category. His candidate for this universal tannakian category is the category of pure motives up to numerical equivalence (which is comletely completely determined by intersection theory in classical Chow groups of smooth and projective varieties over a field).
2) Mixed motives --- But this is only a small part of the story (or, if you wish, of the yoga). Cohomology theories, like l-adic cohomology or de Rham cohomology, are not defined only for smooth and projective varieties, and they don't come alone: they come with a whole bunch a derived categories of coefficients (in our examples, l-adic sheaves and D-modules), which have very strong functoriality properties, reflecting dualities and gluing data (expressing decompositions into a closed a subscheme and its open complement) as well as nice descent properties (mainly étale and proper descent). The idea is that any computation or construction which involves only these functorialities (known as the "6 Grothendieck operations") and which holds in all the known examples should be motivic as well (in particular, intersection theory should appear naturally from there; non trivial structures on cohomology groups, like weight filtrations for instance, should also be explained by these functorialities). I mean that there should exists a theory of motivic sheaves which should be the universal system of coefficients M over schemes (not necessarily over a field). Given another system of coefficients A (like l-adic sheaves) we should get tensor exact functors M(X) --> A(X) (for any scheme X) which are compatible with Grothendieck's 6 operations (i.e. pullbacks, direct images with or without compact support, etc). If the categories A(X) are tannakian, these These realization functors are also conjectured to be faithful. All the regulator maps are expected to come from such realization functors. At last, the category of pure motives mentioned above should be a full tensor subcategory of the abelian category of mixed motives over the ground field.
The existence of such motivic sheaves has been conjectured in some way or another by Grothendieck, Deligne, and Beilinson. However, as they noticed themselves, we can weaken these requirements by replacing the categories of coeffcients by their derived categories D(A), and only require that we have triangulated categories of mixed motives over schemes (without asking that they are derived categories of an abelian category). The good news are then that, if we allow these categories of coefficients to be abstract triangulated categories, then such a universal functorial theory of mixed motives over arbitrary schemes is not completely out of reach: the work of Voevodsky, Suslin, Levine, Morel, Ayoub and al. on homotopy theory of schemes makes it already quite close to us: this theory allows to produce triangulated categories DM(X) such that triangulated tensor functors DM(X) --> D(A)(X) actually exist (and are compatible with Grothendieck's 6 operations), while the Hom's in DM compute exactly higher Chow groups (but we don't know if they are conservative, as expected). Hence a significant part of the Yoga is becoming actual mathematics nowdays, via the homotopy theory of schemes.
