Usually: A glazed or loose serpentine belt slipping on a cold pulley.
But it can also be: A failing tensioner, or a seized accessory the belt is trying to turn.
Squealing when you first start the car
This one is almost always the serpentine belt, the single belt that runs your alternator, power steering pump, and A/C compressor off the front of the engine. When it's cold, or damp from overnight condensation, the rubber gets a little stiff and doesn't grip the pulleys as well for the first few seconds. It slips, and slipping belt makes that high-pitched squeal. Once things warm up and the rubber loosens, the noise usually goes away on its own, sometimes in two or three seconds, sometimes closer to ten.
A belt that's glazed (shiny and hardened from age and heat) or just a little loose is the routine version of this. Less often, the noise is coming from the tensioner, the spring-loaded arm that keeps pressure on the belt, if that's weakening or its bearing is going bad. Rarely, it's one of the things the belt drives, like the A/C compressor or power steering pump, starting to seize up and fighting the belt every time it turns.
You can usually narrow it down yourself. Pop the hood once the car's cold, before you start it, and look at the belt. If it's cracked, glazed shiny, or you can twist it more than about half an inch with your fingers, that's a loose or worn belt and it's the likely cause. If the belt looks fine and reasonably tight, the noise is more likely the tensioner or a pulley bearing, and that's harder to check by eye. A shop can put a bar on the tensioner and feel whether it's still doing its job, or run a squeal down with a mechanic's stethoscope while the engine's cold.
It's safe to keep driving in the short term. A slipping belt doesn't strand you, until it does, either by cracking through or by a tensioner finally letting go, at which point you can lose your alternator and power steering at the same time, which is a bad moment on the road. The thing to watch is duration. If it used to clear in two seconds and now it's going ten or fifteen seconds every morning, or showing up on warm days too, that's the belt or tensioner getting worse, not better, and it's worth getting looked at before it fails on its own schedule instead of yours.
Fixing it is generally straightforward: replace the belt if that's the cause, or the tensioner if it's lost its spring pressure or the bearing's rough. Both are common, well-understood jobs, and cost varies with the vehicle, but it's not an expensive repair as these things go. If it turns out to be a seizing accessory, that's a bigger conversation, but it's also the least likely of the three.