Since Newtonian gravity is analogous to electrostatics shouldn't there be something called negative mass? Also, a moving charge generates electric field, but why doesn't a moving mass generate some other field?
General Relativity is a mathematical model that relates the curvature of spacetime to an object called the stress-energy tensor. In many cases the stress-energy tensor is dominated by mass and you can simply consider the curvature as being related to the mass. However this isn't always true as I'll mention below.
Anyhow, we can put any numbers we want into the stress-energy tensor and then calculate the curvature. If we put in a positive mass we get (in the Newtonian limit) the usual law of gravitation, but we could put in a negative mass and we'd get a repulsion just as you do in electrostatics. Matter with a negative mass is usually referred to as exotic matter and is a favourite trick for building weird objects like the Alcubierre faster than light drive or wormholes.
However just because we an put exotic matter into Einstein's equation doesn't mean it's physically reasonable to do so. No-one has ever observed exotic matter, no-one has ever come up with a convincing theoretical reason for it to exist. So while we can't prove exotic matter doesn't exist few of us think it does - though we'd all love to be able to build a faster than light drive!
Even though we've never observed exotic matter, we have (we think) observed dark energy. This isn't matter, and doesn't have a negative mass, but it does cause a gravitational repulsion.
Moving mass does generate gravitation different from stationary mass. This is the ''gravitomagnetic'' effect predicted by Lens and Thirring in the 20's and measured by Gravity Probe B:
It is related to the ''frame dragging'' effect that you hear about with respect to spinning black holes. There, there is a spin-dependent radius where an observer will be outside the horizon, and able to escape to infinity, but will not be able to be still relative to infinity, even with an infinitely strong rocket--they will be forced to co-rotate with the black hole.
There are three different measures of an object's mass: its inertial mass $m_i$ (defined by Newton's second law), its passive gravitational mass $m_p$ (defined by how much force it feels in a gravitational field), and its active gravitational mass $m_a$ (defined by the strength of the gravitational fields it makes). You get qualitatively different predictions depending on which of these you take to be negative.
An object with negative $m_p$ but positive $m_i$ would fall up. Antimatter, for example, has been verified to high precision to have the same $m_i$ as matter, but historically there have been suggestions that it might have negative $m_p$. Nobody considers this very likely at this point, but experiments are under way to test it empirically, and preliminary results have been reported (Amole 2013).
If the sign of $m_a$ differed from the sign of $m_p$, conservation of momentum would be violated. This would cause serious problems for all fundamental theories of physics, and experiments have put strict limits on nonconservation of momentum (see, e.g., Bartlett 1986).
In relativity, mass and energy are equivalant, so we refer to mass-energy rather than just mass. In general relativity, matter's ability to produce gravitational fields is determined not just by its mass-energy but also by other variables such as pressure. Mathematically, this is encoded into an object called the stress-energy tensor. The measurement of mass-energy density, pressure, etc., also depends on the observer's frame of reference. For these reasons, the concept of positive or negative active gravitational mass is more complicated to describe in relativity than just specifying the sign of a single number. Relativists talk instead about energy conditions.
The weak energy condition (WEC) says that the energy (i.e., mass-energy) density is never negative in any frame. The dominant energy condition (DEC) is like the weak energy condition, but it also guarantees that no observer will see a flux of energy flowing at speeds greater than c. The strong energy condition (SEC) essentially states that gravity is never repulsive.
The SEC is observed to be violated by dark energy. In fact, all the energy conditions are expected to be violated by certain quantum-mechanical systems (Barcelo 2002). However, there is no known form of tangible matter that would violate these energy conditions.
Amole et al., Nature Communications 4, Article number 1785, doi:10.1038/ncomms2787, http://www.nature.com/ncomms/journal/v4/n4/full/ncomms2787.html
Barcelo and Visser, http://arxiv.org/abs/gr-qc/0205066
Bartlett and van Buren, Phys. Rev. Lett. 57 (1986) 21, summarized in Will, http://relativity.livingreviews.org/Articles/lrr-2006-3/