A central force has the form $$\mathbf F(\mathbf r)=F(r),\hat{\mathbf r},$$ so it always points along the line joining a particle to a fixed center. Gravity and electrostatic forces between point particles are central.

Because $\mathbf r\times\mathbf F=0$, angular momentum about the center is conserved. The motion therefore remains in the plane perpendicular to the constant angular-momentum vector.

A two-body problem can also be reduced to an equivalent one-body problem. For masses $m_1$ and $m_2$ interacting through a potential that depends only on their separation $r=|\mathbf r_1-\mathbf r_2|$, separate center-of-mass motion from the relative coordinate $$\mathbf r=\mathbf r_1-\mathbf r_2.$$ The relative motion behaves as a single particle of reduced mass $$\mu=\frac{m_1m_2}{m_1+m_2}$$ moving in the same interaction potential.

For the Earth-Sun system, $m_{\odot}\gg m_E$, so $$\mu\approx m_E,$$ which explains why treating the Sun as fixed gives an excellent first approximation. For two comparable masses, however, both orbit their common center of mass and the reduced-mass formulation is essential.

Central-force reduction turns a six-coordinate two-particle problem into center-of-mass translation plus a planar relative-motion problem. This is the starting point for effective potentials, Kepler orbits, scattering and many quantum two-body systems.