Fluid statics is the field of physics that involves the study of fluids at rest. Because these fluids are not in motion, that means they have achieved a stable equilibrium state, so fluid statics is largely about understanding these fluid equilibrium conditions. When focusing on incompressible fluids (such as liquids) as opposed to compressible fluids (such as most gases), it is sometimes referred to as hydrostatics.
A fluid at rest does not undergo any sheer stress, and only experiences the influence of the normal force of the surrounding fluid (and walls, if in a container), which is the pressure. (More on this below.) This form of equilibrium condition of a fluid is said to be a hydrostatic condition.
Fluids that are not in a hydrostatic condition or at rest, and are therefore in some sort of motion, fall under the other field of fluid mechanics, fluid dynamics.
Major Concepts of Fluid Statics
Sheer stress vs. Normal stress
Consider a cross-sectional slice of a fluid. It is said to experience a sheer stress if it is experiencing a stress that is coplanar, or a stress that points in a direction within the plane. Such a sheer stress, in a liquid, will cause motion within the liquid. Normal stress, on the other hand, is a push into that cross sectional area. If the area is against a wall, such as the side of a beaker, then the cross sectional area of the liquid will exert a force against the wall (perpendicular to the cross section - therefore, not coplanar to it). The liquid exerts a force against the wall and the wall exerts a force back, so there is net force and therefore no change in motion.
The concept of a normal force may be familiar from early in studying physics, because it shows up a lot in working with and analyzing free-body diagrams. When something is sitting still on the ground, it pushes down toward the ground with a force equal to its weight. The ground, in turn, exerts a normal force back on the bottom of the object. It experiences the normal force, but the normal force doesn't result in any motion.
A sheer force would be if someone shoved on the object from the side, which would cause the object to move so long that it can overcome the resistance of friction. A force coplanar within a liquid, though, isn't going to be subject to friction, because there isn't friction between molecules of a fluid. That's part of what makes it a fluid rather than two solids.
But, you say, wouldn't that mean that the cross section is being shoved back into the rest of the fluid? And wouldn't that mean that it moves?
This is an excellent point. That cross-sectional sliver of fluid is being pushed back into the rest of the liquid, but when it does so the rest of the fluid pushes back. If the fluid is incompressible, then this pushing isn't going to move anything anywhere. The fluid is going to push back and everything will stay still. (If compressible, there are other considerations, but let's keep it simple for now.)
All of these tiny cross sections of liquid pushing against each other, and against the walls of the container, represent tiny bits of force, and all of this force results in another important physical property of the fluid: the pressure.
Instead of cross sectional areas, consider the fluid divided up into tiny cubes. Each side of the cube is being pushed on by the surrounding liquid (or the surface of the container, if along the edge) and all of these are normal stresses against those sides. The incompressible fluid within the tiny cube cannot compress (that's what "incompressible" means, after all), so there is no change of pressure within these tiny cubes. The force pressing on one of these tiny cubes will be normal forces that precisely cancel out the forces from the adjacent cube surfaces.
This cancellation of forces in various directions is of the key discoveries in relation to hydrostatic pressure, known as Pascal's Law after the brilliant French physicist and mathematician Blaise Pascal (1623-1662). This means that the pressure at any point is the same in all horizontal directions, and therefore that the change in pressure between two points will be proportional to the difference in height.
Another key concept in understanding fluid statics is the density of the fluid. It figures into the Pascal's Law equation, and each fluid (as well as solids and gases) have densities that can be determined experimentally. Here are a handful of common densities.
Density is the mass per unit volume. Now think about various liquids, all split up into those tiny cubes I mentioned earlier. If each tiny cube is the same size, then differences in density means that tiny cubes with different densities will have different amount of mass in them. A higher-density tiny cube will have more "stuff" in it than a lower-density tiny cube. The higher-density cube will be heavier than the lower-density tiny cube, and will therefore sink in comparison to the lower-density tiny cube.
So if you mix two fluids (or even non-fluids) together, the denser parts will sink that the less dense parts will rise. This is also evident in the principle of buoyancy, that explains how displacement of liquid results in an upward force, if you remember your Archimedes. If you pay attention to the mixing of two fluids while it's happening, such as when you mix oil and water, there'll be a lot of fluid motion, and that would covered by fluid dynamics.
But once the fluid reaches equilibrium, you'll have fluids of different densities that have settled into layers, with the highest density fluid forming the bottom layer, up until you reach the lowest density fluid on the top layer. An example of this is shown on the graphic on this page, where fluids of different types have differentiated themselves into stratified layers based on their relative densities.