Why do my pictures get blurry when I don't use a flash?

I'm using a Nikon D50… No matter how much, or how little, light there's… Every time I turn the flash off, it makes the shutter go really slow. On any setting.

Why? How do I fix this?

Because you have yet to learn how to use your camera! Take a course or buy a book!

Basic principles for beginners:
Turn image stabilization on.
Set the ISO to Auto, 400, or 800 if you are happy with the picture noise at that level
Put the camera into "P"rogram mode, and adjust the shutter speed aperture balance so that the shutter is fast enough to freeze motion.
Learn to brace yourself and hold the camera steady.

*Before anyone criticizes the above remember I'm talking beginner here!

Put it in shutter priority mode, and set your shutter speed to at least 1/focal length.

For example if you use a 200mm lens your shutter needs to be at least 1/200
For a 50mm lens, it only needs to be 1/50

Reading the manual and learning the basics would also help.

Http://www.retrevo.com/search/v2/jsp/downloadPage.jsp?doc=5161048da503233e110c065c1dbc5983&modelid=200550&q=Nikon+D50

Here is the manual. Look up how to set your ISO higher.

It won't do that on shutter priority, so no. Not every setting.

Learn how to shoot in the manual modes to overcome this problem, and learn how aperture, shutterspeed, and ISO affects your exposures. You can read books, take a course, online tutorials, or have a seasoned photographer show you the ropes. And practice.

Have fun

You fix your problem by learning about the Exposure Triangle which consists of Aperture, ISO and Shutter Speed. Go here:
http://www.digital-photography-school.com/learning-exposure-in-digital-photography

Another thing you need to learn is that trying to use the 18-55mm f3.5-5.6G zoom that came with the camera in low-light situations is a serious handicap. Also, your 18-55mm zoom doesn't have VR (Vibration Reduction) which also handicaps you in low-light since the VR allows you to successfully hand-hold the camera at slower than normal shutter speeds. If you plan on a lot of shooting indoors with available light then invest in a 50mm f1.8 or, even better, a 50mm f1.4 prime lens.

Of course if your blurry pictures are being caused by camera shake you need to mount your camera on a tripod. Blurry pictures caused by subject movement can only be avoided by using a faster shutter speed.