Einstein's Equations

Einstein Equation's describe the dynamic of spacetime. They couple geometry to a matter source, i.e., the distribution of matter determines how the spacetime is curved. In turn, the motion of matter is affected by the geometry of spacetime. The situation is similar to electrodynamics where the electric and magnetic fields affect the motion of charged particles which in turn generate electric and magnetic fields. Although one can (formally) write the EE in a very compact form they are in fact very complex and constitute a system of coupled non-linear ellipto-hyperbolic PDE. Even without presence of any matter the solutions of EE can be very interesting and non-trivial (e.g., gravitational waves, black holes). Another  peculiar property  of EEs is the  possibility of  formation  of  singularities from an originally regular matter distribution.
Due to the inherent complexities of the EE, they are exactly solvable only for limited class of problems - typically exhibiting high degree of symmetry and describing simple matter (or no matter at all).