Note: I'm not a power grid engineer. Someone with more detailed information may edit or comment on this answer.
What happens when there is more demand than production?
Then they turn up the production. It can be automatic or manual, as niels nielsen says. Large adjustments (like starting up a power plant) need a human to decide.
The balance of supply and demand is measured by the grid frequency. On average it's 60Hz (in the USA) but there are small fluctuations. If it goes above 60Hz, generators automatically lower their power. If it goes above 60Hz, generators automatically increase their power.
This is because most of the power grid consists of big spinning generators and motors, and they have lots of inertia. If the frequency drops to 59.98Hz, then a motor that's spinning at 59.99Hz is now a generator feeding power back onto the grid, resisting the frequency drop. And vice versa.
What happens if they can't turn up the production? If all the power plants are already running full throttle.
Then the whole system collapses, basically. The frequency goes down and down despite everyone's best efforts to keep it up. At some point, power plants start tripping off (disconnecting from) the grid and doing emergency shutdowns, as the control systems realize they can't keep everything running. That obviously worsens the problem, and within a matter of minutes, everything grinds to a (metaphorical) screeching halt. Everything is dark. No power anywhere.
Who runs out of power?
Now, obviously, the system operator (the company that runs the grid) doesn't want the whole grid to shut down. Partly because it means they're a failure and should feel ashamed of themselves, partly because it's an annoying process to start up the power plants again without any power. So what they will do, before this happens, is make sure there is enough spare production that they could increase production if they needed to. They may start up power plants and let them run idle, for example. If they still don't have enough, they will start disconnecting people from the grid, called a "rolling blackout" or more euphemistically "load shedding". Apparently, that is what you now have in Texas.
What if there's more production than demand? I read that power stations look at historical demand levels to increase production during peak consumption hours, but what if they increase production and everybody turns everything off, driving demand to zero? What happens?
Well they reduce production. Most power plants can shut down, and the ones that can't have to have safety procedures anyway (in case of a trip or emergency shutdown). So they can go all the way down to 0%. If demand decreases faster than they're expecting, they might have to vent hot steam to the atmosphere or something, instead of using it to spin a generator. Wasteful, but not destructive.
When you turn everything back on at once, though, there will be a blackout though and the system operator will be really annoyed.