Long exposure times are an interesting tool that can give your photos a new artistic quality. It does, however, require more time (obviously) and potentially some extra tools to help your camera slow down the shutter speed. I'll try to explain how you can get most any camera to do this to some extent, as long as it gives you the power to control certain aspects of your camera's settings.
How can you take motion in an image and blur it?
Previously, the only way I've been able to slow down my shutter speeds without overexposing the image is by using a couple tricks most any camera should be able to do (as long as it gives you the control to do so) : ISO Settings, and Aperture.
All of these ISO and aperture settings will only allow me to go so far though with longer exposures, since by having to change them I end up affecting some other aspect of my image. Having a low ISO setting (ISO 100 for instance) will actually help with the clarity of the photo while also helping increase exposure time necessary, but setting the lens aperture at its smallest opening (f26 for example) does affect the overall amount of the image in focus, while also deteriorating image quality (through diffraction, too in depth to explain here).
There is one extra tool that you can use for long exposures, though, that also gives you more freedom with other camera settings.
Lens Filters are what really allow you to control shutter speed and give you the capability to really take control of exposure times.
A lens filter is basically a piece of glass you put in front of the lens to affect the amount of light making it to the film/sensor. Some filters help with colors in the image (UV haze), some help with reflections from shiny surfaces (polarizers), and some help with light exposure (neutral density). Neutral Density (ND) filters are designed to help even out the lighting in your pictures, or change the amount of light (and therefor exposure time needed for the image).
I've recently gotten a really dark ND filter, which is basically a near-opaque piece of glass (Lee "Big Stopper" filter). It reduces the amount of light that can make it through the lens drastically, increasing the amount of time necessary for the image to gather light. It allows me to compose an image and choose the aperture that I want to use in order to get the cleanest picture I can (and potentially what I want to be in or out of focus), and still get the motion blur that I like from the longer exposure time.
In order to keep the camera steady for this length of time, I have to use a tripod to hold the camera. I'll compose the image, choosing what I want in the frame and where, and controlling the ISO settings and aperture. Once I do that, I take a few images and check the exposure of the image. Once I've got it dialed in to what I want the exposure to look like, I put on the dark glass filter in front of the lens. The filter is so dark that the camera sensor cannot even tell what is in front of it anymore, so I have to make sure I turn off autofocus (since it will attempt to focus on nothing). I set my camera so that I can control the shutter speed manually, and do the calculations to determine what the new exposure time needs to be in order to achieve the same exposure without the filter. LEE also sends a little card with the Big Stopper Filter that tells you what shutter speed you should use with the filter on vs. off the lens.
By using the filter, I can blur out clouds moving across the sky, waves of the ocean, or any other motion in an image and help focus the viewers attention onto the part of the photo that I wanted to draw their attention to. It allows me to create images that I could not really make without it. Sure, with all of my camera tricks to increase exposure time (ISO and Aperture settings), I can usually change the shutter time from 1/60 second to 1 second. With my dark ND filter, I can change that 1/60 shutter speed to a full minute.