They're both infinite-range interactions that, in the case of a static monopole source, can be described at least approximately by an inverse-square force law. (Exactly in the case of the electric force; for gravity, you get inverse-square only in the limit of large distances.)
In the quantum theory, both are mediated by massless particles (although for gravity, the mediator particle -- the graviton -- probably isn't fundamental).
Electromagnetic waves have 90-degree polarization, and are generated by a changing dipole moment; gravitational waves have 45-degree polarization, and are generated by a changing quadrupole moment. (Spin 1 vector photon vs. spin 2 tensor graviton.) Both waves propagate at the speed of light.
In the weak-field linearized approximation (neglecting gravitational self-interaction), the gravitational field can be decomposed into gravitoelectric and gravitomagnetic fields obeying something analogous to (but not the same as) Maxwell's equations. There is an analog of the Lorentz force law, but the gravitomagnetic force part has a different magnitude and sign. In full GR, this force-based description of gravity breaks down.
Electromagnetic sources have two signs (positive and negative) and the interaction can be attractive or repulsive; gravitational sources have just one (positive) and is always attractive (unless you introduce a cosmological constant or something). Actually, the source of electromagnetism is a vector (4-current) and the source of gravity is a tensor (stress-energy).