Gravity is one of the four known fundamental forces of the universe. It is the one that determines which way is down - in some ways it is the most obvious and familiar of the forces, but it is only since Isaac Newton that people have really thought of it as being a force at all. Before that people just figured that stuff has a natural tendency to fall downwards, which is not quite the same thing; Newton's great insight was to realise that the observed universe starts making a lot more sense if we suppose that everything is attracted to everything else, and that the motions of the moon and the planets and our own Earth around the Sun can be explained by the exact same force that drops an apple to the ground.
Gravity is by far the weakest of the forces. To get an idea of just how weak it is compared to electromagnetism, consider how easy it is to pick up pieces of paper using the static electric charge on a comb that has just been through clean, dry hair: all it takes to overcome the entire gravitational pull of the planet Earth is those few electrons that have jumped from your hair to the comb. Yet gravity dominates at large scales, because electromagnetism has an in-built tendency to cancel itself out - positive charges are so strongly attracted to negative charges that they almost always appear together, and from a distance, the conflicting positive and negative pulls almost always balance each other out.
Gravity is strongly self-reinforcing, since it attracts everything to everything. This gives bulk matter a tendency to congregate and to collapse in on itself, which is one of the main reasons the cosmos orders itself into galaxies, stars, planets and clouds, with vast areas of mostly-empty space between them. Things tend to collapse into balls, on a big enough scale, or spin into discs. When a dust cloud collapses in on itself, its gravitational potential energy is converted into other forms of energy, much of it ending up as heat. This is why the centre of the earth is hot, and how stars get hot and dense enough to initiate nuclear fusion.
Every object in the universe is attracted to every other object in proportion to the mass of each object, and in inverse proportion to the square of the distance between them. That is, an object twice as far away will be attracted one quarter as strongly. This inverse square law is the result of the gravitational field spreading out in all directions from any body. It happens for exactly the same reason that when you look at something from twice the distance, it looks a quarter of the size, in the sense that it is half as wide and half as tall. The area of a sphere (or cube) is therefore in proportion to the square of the distance, and gravity spreads out in a sphere from its source.
At any point in space, the total strength of the gravitational field tells you how strongly it pulls on any unit of mass - the force applied per kilogram of mass, or equivalently the rate at which things accelerate. Right here on the surface of the Earth, everything is pulled downwards at the rate of about 9.8ms-2 (or equivalently, 9.8N/kg) - that is, ignoring drag, any falling object will fall 9.8 metres per second faster (or 22 miles per hour) for every second it falls. We can't ignore drag in the real world, of course, and in practice any given object will eventually reach a terminal velocity where the drag matches the force of gravity. This depends on the weight and shape of the object. For an adult human it is around 55 metres per second - that's more than 120 miles per hour. For a mouse it is less than a tenth of that, so it is probably true that mice can survive a fall from any height - depending on how they land. Cats appear to be slightly more likely to survive after they reach their terminal velocity (which is around about 7 floors down), presumably because they find their feet, stop panicking and very quickly get the hang of being their own feline parachute.
The fact that gravitational field strength can be measured as a rate of acceleration was one of the threads that led Einstein towards the General Theory of Relativity. He started considering what it might mean if gravity is in fact identical to acceleration, in the light of what the Special Theory of Relativity tells us about velocities - in particular, Minkowski's formulation of relativity in terms of spacetime. If velocity can be seen as a rotation in spacetime, and gravity can be seen as a rate of change of velocity, maybe gravity arises from a change in spacetime? Einstein figured out that if he supposed that spacetime is curved by the presence of any mass, he could use the equations of four-dimensional geometry that Riemann had worked out sixty years before, to produce something that looked very much like Newton's equations of gravity... until you looked at what happens near extremely massive bodies, or at subtleties like the way light curves around stars.
Einstein's conception of gravity has a number of very interesting consequences. One is that the distortion of spacetime around every massive body means that the closer you get to it, the more time slows down - so time passes measurably slower for us than it does for a satellite in orbit. The effect is small here, but around much more massive bodies, it would become far more noticeable.
Another rather odd consequence of General Relativity is that from a certain point of view, gravity is not a force at all - it is just what happens when things follow their natural trajectory along the most direct path through spacetime. Indeed, someone in freefall does not feel anything like a force of gravity - which is why astronauts in orbit feel weightless, although they have by no means escaped the pull of the Earth's gravity entirely. From the more familiar frame of reference of someone standing on the surface of a planet, of course, gravity looks very much like a real force - just as it makes sense for someone on a fast-spinning roundabout to treat centrifugal force as a real thing. Indeed, there are two ways of viewing the weightlessness of orbit. One is that being in orbit is like being in freefall, while moving sideways so quickly that you never touch down. The other perspective is that since you are moving around in an ellipse at just the right speed, you feel a centrifugal force that exactly matches the pull of gravity, cancelling it out. The distinction between 'fictitious' and 'real' forces is not clear-cut, and string theory suggests that all forces may in fact depend on one's frame of reference - which would make them, in some sense, fictitious.
Black holes and wormholes might also be fictitious, and they certainly sound unlikely when you first hear about them. In fact, though, black holes are almost certainly quite real, and wormholes are at the very least plausible. There is no apparent limit to the amount of spacetime curvature that general relativity allows, you see, and no known force in the universe which can resist a strong enough gravitational pull. That makes it seemingly inevitable that when sufficient mass collects in one place, it will eventually collapse into a gravitational singularity - a point, or ring, of infinite spacetime curvature, from which almost nothing can escape. Although we cannot observe it directly - conclusively proving the existence of an actual black hole is extremely difficult, even in principle - the signs are very strong that there is a black hole at the centre of our own galaxy, with the mass of more than four million suns. There is less reason to think that our universe contains actual wormholes, which is to say hyper-dimensional tunnels connecting two regions arbitrarily far apart in space and in time. However, they are at the very least a tantalising possibility suggested by the mutability of spacetime - nobody has been able to rule out that it might be possible to build one.
In daily life, of course, we usually don't have to think about relativistic gravity - in fact, as long as we stay close to the surface of the Earth, even Newton's universal gravitation is barely relevant. The main thing to know on Earth is that the gravity around here accelerates things downwards at about 9.8ms-2, which means that a 1kg weight gains 9.8J of kinetic energy for every metre it falls (or gains the same amount of potential energy for every metre you raise it up). Having said that, the effective strength of gravity on Earth actually varies from about 9.832ms-2 at the poles to about 9.780ms-2 at the Equator, so a 1000lb pumpkin at the North Pole would only be a 994.7lb pumpkin by the time you got it to the Equator. Even for interplanetary space missions, good old-fashioned Newtonian physics is generally enough to plan your trajectories around the solar system. It is true, though, that as soon as you are in orbit you will start gaining 38 microseconds on us every day. Granted, that is only one full second every 73 years - but 38 microseconds of time is also the equivalent of seven miles' worth of space, and sometimes that is important.
Further reading
Richard Feynman's Lectures on Physics are excellent on gravitation (see chapter 7 of volume one). You could read that for free on the internet, but that would be illegal. Most of this stuff is remembered from my undergraduate days; any time I was at all unsure I checked against various Wikipedia articles, whatever showed up in Google and/or people who sort of know what they're talking about. The stuff about falling animals turns out to be quite difficult to get hard figures on. The How Far Can An Animal Fall And Still Survive? node was handy, and this demonstration from the BBC was fairly persuasive, while this page has someone attesting that falling mice aren't always so lucky, and here is a source for the report that cats tend to fare a bit better if they fall even further than seven floors. Without this page I would probably have included the extremely plausible myth that the Global Positioning System wouldn't work at all without relativity.