Assuming you're in aperture prioirity and not manual it sounds like you're setting the aperture too small. A small hole only lets in a little bit of light so the shutter has to remain open longer to achieve the correct exposure.
Small number means big hole, means lots of light, means short shutter duration so you have less chance of camera shake being evident in your pictures.
Big number means little hole, means less light, means longer shutter duration and the greater chance of camera shake unless you're using a tripod.
Ignoring the effects the aperture has over the depth of field if you're opening the aperture right up to let lots of light in and you're still getting camera shake from long exposures you either need to bring more light into the picture or turn up the ISO setting.
I could rattle on for ages but I won't yet. Hopefully this is enough to start with, just shout if you want more.