the stress-energy tensor
A single particle carries energy and momentum as a four-vector. But a continuous system — a fluid, an electromagnetic field, a gas — has energy and momentum spread through space and flowing from place to place. The stress-energy tensor is the object that keeps track of all of it: how much energy and momentum sit at each point, and how they flow. It is the source of gravity in general relativity.
The stress-energy tensor T^munu is a symmetric rank-2 four-tensor whose components are densities and fluxes: T^00 is the energy density; T^0i is the density of the i-th momentum (equivalently the energy flux over c); T^i0 the flux of energy (equal to T^0i by symmetry); and the spatial block T^ij is the flux of i-momentum in the j-direction — i.e. the stresses (T^ii are pressures, off-diagonal are shear stresses). Its defining physical law is the local conservation of energy and momentum, written covariantly as partial_mu T^munu = 0 — four continuity equations in one, generalizing partial_mu J^mu = 0 for charge.
You meet T^munu everywhere continuous energy-momentum appears. For a perfect fluid at rest it is simply diagonal, T^munu = diag(rho c^2, p, p, p) — energy density in the time-time slot, pressure p on each spatial diagonal — and it is boosted to moving frames using the fluid four-velocity. For the electromagnetic field it is built from F^munu and contains the electromagnetic energy density, the Poynting vector (energy flux), and the Maxwell stress tensor as its blocks. Above all it is the source in Einstein's field equations G^munu = (8 pi G/c^4) T^munu — energy and momentum tell spacetime how to curve. Caveat: partial_mu T^munu = 0 expresses conservation only in flat spacetime; in curved spacetime the ordinary derivative is replaced by the covariant derivative, nabla_mu T^munu = 0, which is local (not global) conservation because energy can be exchanged with the gravitational field.
For ordinary matter at rest the dominant component is T^00 = the energy density = rho c^2; a static electromagnetic energy density u sits in T^00 while the Poynting vector S/c fills T^0i, showing momentum flow even when nothing has mass.
One symmetric 4x4 tensor holds energy density, momentum density, energy flux, and stress together.
partial_mu T^munu = 0 is local conservation; it does NOT guarantee a globally conserved total energy in curved spacetime, because gravity itself carries energy that T^munu (matter plus fields) does not include. The symmetry T^munu = T^nu mu is tied to the conservation of angular momentum.