The first part of the trinity that controls your exposure, shutter speed is pretty simple. It's just how long the shutter is open for. If it's open for longer, more light gets in, the image is brighter. Shorter time, less light, image is darker.
Combining with Aperture and ISO
Change the shutter speed by making it faster, and open the aperture, making it wider, and the result is the same exposure. The image will be the same brightness.
When you're starting out, leave the camera on auto, and don't worry about it.
When you start to care a bit more, if you're anything like me, you'll care more about the aperture, and the shutter speed will just be there to balance it. But if you don't care about it at all, you'll soon end up with blurred images.
Getting the Right Shutter Speed to Freeze the Action
Unless you're aiming for motion blur, you usually want to freeze the action, which means a shutter speed that's fast enough. How fast that is depends on what you're shooting. Sports usually need pretty fast shutter speeds. Even with modelling, sometimes you want to freeze the action to catch the model jumping or flinging their hair about or splashing in the sea. All the things that make a shoot more fun.
For most reasonably normal shooting situations, without a particularly long lens on the camera, 1/125th is probably ok. From 1/500th or so you'll freeze most normal movement. If things are really moving fast, 1/1000th or even 1/2000th can be good.
The Numbers
1/125th? What? Of a second. Shutter speed is measured in fractions of a second. Unless it gets very slow, when it starts being measured in seconds, but at that point you're getting a bit specialist, and you need a decent tripod.
The numbers are generally just halving each time, but with a bit of rounding to keep the numbers easier. Starting at 1 second, we then go to 1/2, 1/4, 1/8, 1/16, 1/30, 1/60, 1/125, 1/250, 1/500, 1/1000, 1/2000, 1/4000, 1/8000.
See the jumps? After 1/16th of a second really should have been 1/32nd of a second. But there's so little difference, someone decided not to care, and just rounded it. I like them - why complicate things for the sake of 0.06 seconds? After 1/60th (which should really have been 1/64th) should have been 1/120th, or maybe 1/128th. Again, it got rounded to 1/125th, which then makes the rest of the numbers nice and easy.
When I was younger, most cameras could get to maybe 1/1000th of a second, but they got faster. My current camera (Sony Alpha 7iii) can do 1/4000th, and 1/8000th is becoming fairly common.
The Need for Speed
Such high shutter speeds are rarely needed to freeze the action - 1/1000th is usually more than enough. And they aren't needed to get the right exposure, because you could just close down the aperture instead. But I'd find a maximum of 1/8000th to be pretty useful. Why? Because I usually don't want to close the aperture down.
As you'll read about in the next section, a wide aperture can give a nice look to your photos. And it's a look I love. So I want to use a wide aperture, even in bright light, and on a sunny day, at f/1.4 (if that means nothing to you, we'll cover it soon) my camera will start warning me it's over-exposing because it can't use a fast enough shutter speed.
There are other ways around it, like using a Neutral Density Filter to block out some of the light, but a faster shutter speed is easier.
How I Handle Shutter Speed
I live in Aperture Priority mode. I set the aperture on the lens, and let the camera decide what to do with the shutter speed. Sometimes I change the exposure a bit using the exposure compensation dial. I find it an easy way to work, letting the camera deal with things as they change, and I'm rarely too worried about the shutter speed unless the light is getting low.
I usually also have my camera set to Auto ISO - with this enabled, if it gets too dark, the camera will increase the ISO setting to keep the shutter speed fast enough to avoid too much risk of camera shake. With many cameras now, this is done quite intelligently. Because I'm using manual lenses on my Alpha 7, though it's pretty dim about it. It'll only increase the aperture if the shutter speed would get below 1/60th. With my 40mm lens on the camera, that's just about good enough, but not ideal. If I'm using, say, my 90mm lens, it's not enough, and I'll get camera shake.
I still think it's better to leave it on, because I'd have more chance at 1/60th than if it had dropped below that, but I have to keep an eye on what it's doing - if the ISO starts jumping up, the shutter speed is already too low, and I'll have to intervene. A quick look at what the ISO is being set to, and set it a bit higher.
The problem then is that the images get noisy. We'll talk about that a bit more in the section on ISO, but I usually just give up at that point. I don't really want the noisy images. If you find the images from your camera good enough at higher ISO, keep going. They're getting better all the time.