You may have heard in passing phrases like "space is curved" or "the curvature of spacetime", but what do these phrases actually mean? Einstein's theory of general relativity brought us the understanding that the force of gravity is the result of masses deforming (curving) the fabric of spacetime surrounding them. This is different to the other forces of nature which are quantised and mediated by force-carrier particles, and can be mathematically combined into a single force known as electroweak. Don't worry if that last sentence didn't make sense. The important thing is that our deepest understanding of gravity does not include quantum theories* but does involve classical geometry.
The most common metaphor used to describe the curvature of spacetime gives something like this:
Imagine the universe is an infinite rubber sheet (infinite because this is not the place to think about what's happening at the edges or even whether edges exist). Masses such as stars, planets and so forth are like ball-bearings stuck to the surface which, because of their mass, create dips in the sheet. Heavier things make deeper dips. Then, if you roll another ball bearing along the sheet (it doesn't actually have to be a lighter one, but it's easier to picture if it is). As it rolls past the other masses, it's path will be deflected by the dips so that it doesn't go in a straight line from the point of view of an external observer. If you were to draw a grid on the sheet before letting the masses stretch it, then rolled a very small mass very quickly along along it, it would follow the grid lines, even though the grid lines themselves are now stretched.
The very small, light mass I mentioned at the end would have to be a photon, a massless particle of light. Anything with a mass, even a small one, would get deflected off the grid lines by the masses because it would be travelling more slowly that the speed of light.
Of course, the universe isn't a two-dimensional rubber sheet and this metaphor isn't perfect. It's a little bit harder to picture in three dimensions, but the qualitative ideas are the same. And you can think of the grid lines as geodesics which mean they represent the shortest distance between two points and hence the path light takes. That's right, gravity curves the path light takes. The larger the mass, the greater the curvature. This is called gravitational lensing and is all sorts of useful in different areas of astrophysics.
Now, Newtonian gravity doesn't take the curvature of spacetime into account. Usually this doesn't matter much because even the sun only causes enough curvature for Mercury, the innermost planet, to really be effected. The general relativistic (GR) corrections for the other planets are small enough to be insignificant. GR becomes much more relevant around small and dense objects such as neutron stars and black holes.
* miscellaneous proposed but untested theories notwithstanding.
Black as black
Black holes are an interesting concept that falls out of general relativity. Chances are you've heard of them if you haven't lived in an internetless cave for the past hundred years. But what exactly are they?
A black hole is an extremely small and extremely massive object. It is so small and massive that it is denser than any form of matter we know of. We don't really know what sort of matter black holes are made of. This comes from the fact that we don't have a theory of quantum gravity as I mentioned earlier.
Escape velocity is the speed you need to go to escape a body's gravitational pull. To completely escape Earth's gravitational field, for example, you need to leave the Earth at about 11 km/s which is about forty thousand kilometres per hour. To escape the sun's gravity from the surface of the sun (let's ignore the fact that you'd be fried while you were there) you need to leave at about 618 km/s or more than two million kilometres per hour. To escape the sun's gravity from a distance of 1 AU (the distance between Earth and sun) you need a velocity of 42 km/s (150 thousand km/h) because the force of gravity drops off with the square of distance.
Because black holes are so dense, there is a point where the force of gravity is so strong that the escape velocity is equal to the speed of light. This is called the event horizon and from within the escape horizon nothing can escape the gravitational pull of the black hole (since nothing can move faster than the speed of light). This is where the "black" part of "black hole" comes from.
In practice, it would be very hard to escape the black hole long before you reached the event horizon, purely due to the energy needed to overcome gravity. And even before you reached the event horizon (also known as the Schwarzschild radius), lots of strange and interesting things start happening. Any signals you try to send out (to people further away from the black hole) would be redshifted as the wavelengths of light get "stretched out" by the extreme curvature of spacetime around the black hole. Notice how "spacetime" includes the word "time" as well as "space"? Time also gets stretched out near a black hole and passes more slowly (for complicated reasons I might explain in a future blog post). Not that you would necessarily notice if you were falling into a black hole because the tidal forces would be ripping you apart.
Tidal forces come from the difference in the force of gravity between the end of you/your spaceship closest to the black hole and the end further away. Gravity acts more strongly on the closer end, causing interesting (and painful) things to happen. This is the same principle which leads to moons being tidally locked in their orbits around their planets, but taken to an extreme scale. If the force of gravity on your feet is appreciably stronger than on your head, nothing pleasant will result from your feet being sucked into the black hole more quickly than your head.
In terms of where we can find black holes, that's a good question. The only black hole whose existence we're sure of is the supermassive black hole at the centre of our galaxy. We're also quite confident that most other galaxies also contain central black holes. The supermassive part relates to the fact that they are hundreds of thousands times more massive than the sun. In face, our supermassive black hole, Sagittarius A, is about four hundred thousand times the mass of the sun.
Theoretically, there should also be much smaller black holes around, only a few times the mass of the sun, ranging up to a hundred or so solar masses. These would come from very large stars which reached the end of their lifetimes, went up in a supernova and then collapsed into a (stellar) black hole. Even though there's no reason for these to not exist, we have yet to decisively detect any. The black part of the black hole makes that a little tricky. (The is some radiation coming off them, known as Hawking radiation, as well as some interesting things happing as things fall into black holes, which gives us some candidates, but as far as I know they're still only candidates.)
Barely scratching the surface
So that, very briefly, is what general relativity and black holes are about. You might have noticed that there weren't any equations in this post. (Gasp!) That is because most of maths that describes general relativity requires a maths or physics degree to understand. In my experience, general relativity is usually a graduate level subject (or Honours at most Australian universities).
Maths aside, I feel like I've barely scratched the surface of black holes [insert bad pun here], so I suspect another post focussing on black holes will happen some time in the future.