exchemist:

Four-vectors in GR are objects that transform in a particular way (using the Lorentz transformations) when you change reference frames. They are constructed so as to maintain the form of familiar definitions of quantities like $\vec{p}=m\vec{v}$. To run with the example of momentum, the problem is that the Newtonian 3-momentum is not conserved in collisions when you change reference frames. Therefore, we need to find some kind of similar quantity that *is* properly conserved when we change frames. What we end up with is 4-momentum, which happens to be $(c, v_x, v_y, v_z)$.

4-force is defined as $\bf{F}=\frac{d\bf{P}}{d\tau}$, that is the rate of change of the 4-momentum with respect to the proper time. In GR, the derivative is actually a covariant derivative.