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Lienard-Wiechert potentials give exact potentials and fields of a point charge on arbitrary trajectory. They reveal a moving charge produces velocity fields (∝ 1/r²) and acceleration fields (∝ 1/r). The latter dominates at large distances and is responsible for radiation.
You already know that electromagnetic signals travel at c, and that retarded potentials encode this by saying the potential at position r at time t depends on where charges were at the earlier retarded time t_ret = t − |r − r(t_ret)|/c — the time when the signal that reaches you now was emitted. The Lienard-Wiechert potentials are retarded potentials applied to a point charge moving on a specific trajectory r_s(t). They give exact, closed-form expressions: V = (q/4πε₀) · 1/(κR) and A⃗ = (μ₀q/4π) · v⃗/(κR), where R is the distance from retarded position to field point, v⃗ is the velocity at retarded time, and κ = 1 − (v⃗·R̂)/c is a critical factor encoding the "headlight effect."
The factor κ in the denominator is not a small correction — it qualitatively changes the field structure. A charge moving toward you has κ < 1, so the potential is amplified; a charge moving away has κ > 1, suppressed. This relativistic beaming concentrates fields in the forward direction of a fast-moving charge, which is why synchrotron radiation is beamed sharply forward. Computing the actual E⃗ and B⃗ fields from these potentials (via the usual −∇V − ∂A⃗/∂t and ∇ × A⃗) reveals two distinct contributions of very different character.
The velocity fields (also called "bound fields") fall off as 1/r². They point along the retarded direction modified by the particle's motion, and they look like a distorted Coulomb field dragged along with the charge. Because they fall off as 1/r², the energy flux they carry (∝ E² ∝ 1/r⁴) integrates to zero over a large sphere — they carry no energy to infinity. A uniformly moving charge has only velocity fields: it does not radiate, consistent with the equivalence principle.
The acceleration fields (or "radiation fields") fall off as 1/r. They arise only when the charge is accelerating — a⃗ ≠ 0 at the retarded time — and they are proportional to a⃗ projected perpendicular to the observation direction. Because they fall off as 1/r, their energy flux ∝ 1/r² integrates over a large sphere to a finite, nonzero value: energy escapes to infinity. This is radiation. The Lienard-Wiechert fields are the exact classical solution from which Larmor's formula, synchrotron radiation, and all other classical radiation results can be derived by integration.