the holonomy representation
When you carry a (G,X)-structure's coordinates around a loop in your manifold and return to where you started, the coordinate frame need not come back to itself — it comes back twisted by some symmetry in G. A flat torus loop comes back translated; a loop around a hyperbolic surface's handle comes back transformed by an isometry of H^2. The holonomy representation is the bookkeeping that records, for every loop class, exactly which element of G you pick up. It is the structure's global 'monodromy.'
Precisely, fix a developing map D: M-tilde -> X. For each deck transformation gamma in pi_1(M), the composition D after gamma is another developing map for the same structure, so it equals g . D for a unique g in G; set rho(gamma) = g. The assignment rho: pi_1(M) -> G is a group homomorphism — the holonomy representation — well defined up to conjugation in G (changing the developing map by h in G conjugates rho by h). Its image rho(pi_1(M)) is the holonomy group, a subgroup of G; the pair (D, rho) determines the (G,X)-structure completely.
Holonomy is the bridge from geometry to algebra and is the central coordinate on deformation spaces. For complete structures the holonomy is faithful with discrete image, and M is exactly X / rho(pi_1(M)); for hyperbolic surfaces, varying the holonomy representation in the character variety Hom(pi_1, PSL(2,R)) / conjugation traces out Teichmüller space (one chosen component of discrete faithful representations). Mostow rigidity, in this language, says that in dimension ≥3 a discrete faithful holonomy into PSL(2,C) is rigid: the topology of M alone pins rho down up to conjugacy.
A complete hyperbolic structure on a genus-2 surface S has holonomy rho: pi_1(S) -> PSL(2,R), the orientation-preserving isometries of H^2. The image is a discrete, torsion-free, cocompact subgroup (a Fuchsian group) and S = H^2 / rho(pi_1(S)). Each of the four standard generators a1, b1, a2, b2 maps to a hyperbolic isometry (a matrix with trace exceeding 2 in absolute value), and they satisfy the single surface relation [a1,b1][a2,b2] = 1 inside PSL(2,R).
Genus-2 holonomy: generators of pi_1 land in PSL(2,R) as hyperbolic isometries satisfying the surface relation.
Holonomy is only defined up to conjugacy, so never speak of 'the' element a loop maps to without fixing the developing map. And holonomy alone does not always recover the structure: distinct (G,X)-structures can share the same holonomy representation (the holonomy map from deformation space to the character variety is a local homeomorphism, not globally injective).