I hate using the flash when taking photos. I set a fast shutter speed and played with the aperture settings. It's fine with the flash on and takes the photo really quickly, but when I turn it off it always comes out blurred and I don't understand why. Frustration, please help.
The camera needs a specific amount of light to take a properly exposed picture. How much light hits the sensor is tied to two items: shutter speed and aperture. Shutter speed defines how long light hits the sensor and aperture defines how much light hits per unit of time.
If you want to avoid blurry images you have to set a fast shutter speed as you've done, but that means that to take a properly exposed image, you need to adjust other things to make up for less light hitting the sensor. You can open up the aperture, but a lens has limits, and if you need more light than a wide open aperture can give for a shutter speed, the image won't work. That's physics.
Flash is used to add more light to a low-light situation so that the amount of light that hits the sensor is adequate. If you can't use flash and you can't adjust the shutter speed and aperture to get an adequate amount of light, your options are limited to playing with your ISO.
ISO defines how sensitive the sensor is to that light. If you raise the ISO, then the sensor is more sensitive to light. The side effect of this is that the sensor will generate more noise into the image, especially in the dark areas. If you raise the ISO too high, the noise will make the image unusable -- what that means is a judgement call, so you need to experiment here.
But unfortunately sometimes you simply can't get around the light and you need to decide what compromise to give up on -- do you accept the blur? do you accept the noise? Or do you strap on a flash? Part of figuring out photography is learning how to adjust for these factors to get the image you want.