the developing map
Suppose you walk around a manifold carrying a (G,X)-structure, and at each step you write down where you are inside the model space X using the local chart. As long as you stay in one chart this is unambiguous, but to compare distant places you must analytically continue the charts along a path, gluing each new chart to the previous one by the transition element of G. The developing map is the global record of this continuation: starting from a basepoint, it follows every path on the universal cover and assigns to it a single, consistent location in X.
Precisely, let M-tilde be the universal cover of M carrying the lifted (G,X)-structure. The developing map is a map D: M-tilde -> X that is a local diffeomorphism (in fact an immersion compatible with the charts) and is unique up to post-composition by an element of G. You build it by analytic continuation: fix one chart near the basepoint, then for any other point follow a path from the basepoint, and at each overlap correct the next chart by the transition element so the pieces fit together into one map into X. Because M-tilde is simply connected, the result is independent of the path chosen, so D is well defined on all of M-tilde — but it need NOT descend to M, since going around a nontrivial loop changes D by a fixed element of G.
That failure to descend is precisely the holonomy: D(gamma . x) = rho(gamma) . D(x) for every deck transformation gamma, where rho is the holonomy representation. So the developing map and holonomy are two faces of one object. The structure is complete exactly when D is a covering map onto X (then M-tilde = X and M is a quotient of X by a discrete subgroup of G), and incomplete when D is, say, an open embedding onto a proper subset — a hyperbolic structure on a surface-with-cusp, viewed the wrong way, can develop into only part of H^2.
Take a Euclidean cone of cone angle 2pi/3 (a flat disc with a 120-degree wedge cut out and the edges glued). Away from the cone point it has a flat (G,X = R^2)-structure. Develop a small loop encircling the cone point: as you go once around the loop, the developing map sweeps out only a 120-degree sector of R^2 before the edges identify, so D is a local diffeo near each point but globally a 'spiral staircase' that fails to close up. The holonomy of the loop is the rotation by 120 degrees that glues the sector's two edges.
Developing a loop around a cone point sweeps a wedge of R^2; the holonomy is the rotation closing that wedge.
The developing map is an immersion (local diffeomorphism), not generally an embedding: it can overlap itself, and for an incomplete structure it may miss part of X entirely. 'Developable' here means 'can be unrolled by analytic continuation,' which is automatic for any (G,X)-structure — it is not an extra hypothesis.