I'm not someone with good scientific knowledge, so if my question are weird, correct me.
General relativity is expressed in a set of differential equations. The solutions of these equations depend on the initial assumptions of the problem under consideration and the boundary conditions it imposes.
The Big Bang is a particular solution of general relativity equations that is believed to obey the boundary conditions and the observations of astrophysics about the early universe:
The Big Ban is a specific cosmological model within the framework of general relativity.
It's not possible to give a very satisfactory answer without going into a lot of detail about what General Relativity is and how it works. But that would take a book rather than a few paragraphs, so this is a very rough description.
In General Relativity the Einstein equation relates the curvature of space to how matter (and energy) is spread about. For example, if you take a chunk of matter concentrated into a point and solve Einstein's equation you get the solution that describes a black hole.
If we look around the universe we find that on large scales the distribution of matter is roughly the same everywhere. Obviously on a small scale matter is concentrated into stars, then galaxies, then galaxy clusters, but if you go to a large enough scale you find the distribution of galaxy clusters roughly evens out.
So if we assume that matter is evenly distributed in the universe and feed this into the Einstein equation what solution do we get? Well we get a solution called the FLRW metric (the link contains lots of info about this but it's probably a bit technical for non-nerds), and there is general agreement amongst scientists that the FLRW metric seems a pretty good description of our universe.
But the FLRW metric predicts something a bit odd. It predicts that space is expanding. Let me try and make it clear what I mean by this. Suppose you take two small objects like ping pong balls, put them a light year apart and make sure nothing is acting on them, no gravity and no electrical or other forces, then you would expect the ping pong balls to just sit there forever. But what the FLRW metric tells you is that if you wait about 14 billion years the ping pong balls will be two light years apart. This is because the space in between them is expanding: the balls haven't moved, it's the space between that has stretched.
So we know that if we wait the ping pong balls will move apart, but what happens if we wind time backwards? Well the universe is about 14 billion years old. If we wind time back by half that amount, 7 billion years, we would find that the distance between the ping pong balls has reduced to half a light year. If we wind time back to 0.01% of 14 billion years the distance between the balls will be 0.01% of a light year, and so on. If we wind time back to zero we find the distance between the ping pong balls is zero. This is the point we call the Big Bang.
But the Big Bang is stranger than the paragraph above suggests. I've described two ping pong balls, but suppose we have a grid of ping pong balls spaced at one light year intervals and spreading out forever in all directions. If the idea of a grid of ping pong balls seems rather forced let's say it's a grid of stars. After all, as far as we can see stars spread out in all directions forever (admittedly grouped into galaxies, but that's a detail). Now start winding time backwards towards the Big Bang, and the initial one light year spacing in our grid decreases. Wind time back to zero, i.e. the moment of the Big Bang, and we have the strange conclusion that the spacing between all the ping pong balls/stars in the universe is zero, so the distance between any two randomly selected stars is zero: everything in the infinite universe is in the same place! This is why you'll hear it said that the Big Bang happened everywhere. It wasn't an explosion like a bomb going off.
Even we crazy scientists don't believe that everything in the universe could be in the same place, and we describe the Big Bang as a singularity. This is a point where physical quantities become infinite and we can no longer do the maths. For example, try to calculate the size of the universe at the Big Bang. Well we know our grid of ping pong balls/stars goes on forever, but at the Big Bang the grid spacing is zero. This means the size of the universe at the Big Bang is zero times infinity, and this doesn't make physical sense. What it means is that General Relativity can't tell us what happened at the Big Bang. We expect that some more complicated theory, like String Theory, will be needed for this.