Yesterday was Aperture, the day before that it was what camera to use, and today we're going to talk about Shutter Speed. What do all these things have in common.....LIGHT! Just so that we're on the same page I'm going to say it again the one thing you have to have in order to make a photo is light. It is the be all, end all of your photography without it you have nothing. But what does shutter speed have to do with light and how does it work with aperture. Shutter speed is exactly what it's name implies. It is the speed in which your shutter moves. The faster the movement the faster the shutter speed has to be. When you hear your camera make that click sound you're hearing your shutter open and close. The longer it stays open the more light that is going to enter your camera just like curtains to a room. Close it fast and less light will enter. Shutter Speed works in seconds meaning if you set your shutter speed to 1 it will take one second to open and close. That doesn't mean that if you set it to 3 it would then stay open for 3 seconds because with shutter speed the numbers go backwards. The higher the number the less time your shutter stays open working in 1000th of a second intervals. so at 3 you are shuttering 3/1000th of a second and so on. While aperture controls the depth of field in a picture (how much is in focus) shutter speed measures how much motion is captured in a picture. In a way you can think about in how your eyes work. Our eyes blink constantly to allow light to enter and for our brain to process what its just seen. If you open your eyes as someone is dancing around in front of you but don't blink the image will become blurred because too much light is going in and not enough time is being allowed to process what its seeing. Try it and you'll feel your eyes want to close so that they can tell the brain what they're seeing and the person dancing in front of you will be a blur around the room, but let your eyes blink normal and you'll feel them blink fast and faster as the dancer moves. The same goes for the Shutter in your camera. The shutter wants to tell your sensor what it is seeing so that it can then make the image for you (really your camera just wants to make you happy). So the higher the Shutter Speed the more action you will be able to capture.

(sorry for the quality of this image without my own computer I had to up load this from a smaller image from my papa's laptop.)

See how Big Bird and her mighty opponent seem to freeze right in the spot the picture what taken. That is what a higher shutter speed will do. It freezes motion. So the faster and image moves the higher your Shutter Speed has to be, but the less light that will get through. This will limit were and when you can use a higher shutter speed. A lower Shutter speed 30/1000th of a second to 1 second is the ideal speeds to use in an indoor setting HOWEVER This will limit the amount of motion you can capture, because as the number goes down and the longer the shutter is open the more your camera is going to feel. The simplest movement of a subject or a shake of your hand no matter how slight will affect the out come of your photo. A simple rule of thumb used is that your shutter speed should never go less then the length you are shooting at to avoid "Camera shake" meaning if you are shooting at 50 mm than your Shutter Speed should be no less than 50/1000th of second.
All these thing come into play with Aperture and how these two things work together we will cover next. My hope is that I can help you reach a point where you are taking pictures inside without a flash because personally and this is just my personal view on the subject of flash so please people who love flash and the people who makes flash don't send me hate mail. But flash in my mind is the devil but we will save that for another time.
So to recap:
Shutter Speed controls the amount of motion you can capture in a photograph. The faster the shutter speed the more motion you will be able to freeze but with less light hitting your camera's sensor.