the Einstein tensor
Einstein needed a specific quantity built from curvature to place on one side of his field equations, opposite the matter. It could not be just any combination: to match a matter content that never creates or destroys energy, the geometric side had to be automatically conservation-respecting, no matter what. The Einstein tensor is exactly the combination of curvature engineered to have that property built in. It is curvature packaged so that it is ready to be set equal to matter.
It is defined as G_ab = R_ab - (1/2) R g_ab, the Ricci tensor minus half the Ricci scalar times the metric. Its defining virtue follows from the contracted Bianchi identity: its covariant divergence vanishes identically, nabla^a G_ab = 0, for any metric whatsoever. This is a mathematical fact about geometry, true before any physics is imposed. It is also, up to an overall constant and an added multiple of the metric (the cosmological-constant term), the unique symmetric tensor with vanishing divergence that can be built from the metric and its first two derivatives.
You meet the Einstein tensor as the entire left-hand side of the field equations, G_ab = (8 pi G / c^4) T_ab. Its automatic divergencelessness is not a technicality; it forces the right-hand side to obey nabla^a T_ab = 0, which is the local conservation of energy and momentum. Geometry, in other words, guarantees the conservation law, rather than the two being imposed separately. The caveat worth stressing is that the trace of the Einstein tensor is minus the Ricci scalar in four dimensions, G = -R, so it repackages, rather than adds to, the curvature information already in R_ab.
Think of the Einstein tensor as an accountant's ledger for curvature: whatever curvature the matter deposits, the ledger is arranged so the books always balance (nabla^a G_ab = 0), which is why energy and momentum can never leak into or out of spacetime unaccounted for.
Built-in conservation: the geometry's books always balance.
The Einstein tensor is not an independent new curvature; it is a fixed linear combination of the Ricci tensor and Ricci scalar. Its special role is entirely due to its identically vanishing divergence, which no other simple curvature combination shares.