Tilt-shift photography originally referred to the movements on a view camera to focus and control perspective. An example of this is when you are photographing a building and the building looks like it may be leaning or falling in the image. Using tilt-shift controls is a way to correct the distortion or perspective of the lines in the frame and object. In view cameras, the movements are controlled both in the front of the camera near the lens, and in the back near the film.

Starting in the 1960s, camera manufacturers started producing lenses or adapters for 35mm cameras that embodied the same principles. These lenses are often used in architectural photography to control perspective (like mentioned above), and in landscape photography to get an entire frame sharp. In more modern times, photographers have begun using tilt-shift lenses for selective focusing purposes in portrait work, fine art, and miniature scene simulation photography.

The two different types of movement in tilt-shift photography are tilt, where the rotation of the lens plane is relative to the image plane, and shift, where the movement of the lens is parallel to the image plane. Tilt controls the focus or part of the image that appears sharp, while shift is used to adjust the position of the subject in the frame without moving the actual camera.

There are a few ways to achieve a similar look to tilt-shift, which is what we'll be showing through examples below. (The first of course, is actually shooting with a view camera).

Image (above) taken with normal 50mm lens, slightly shallow depth of focus
Option: The Tilt-shift lens
 Image taken with a tilt-shift lens for SLR cameras
A tilt-shift lens. This one can be used on a 35mm film EOS camera, or a digital SLR camera. 
The tilt and shift are moved by turning the two knobs on the sides of the lens, as well as by rotating the base of the lens. The focus is done by hand on the front focus ring. It's important to keep in mind that even though you may mount a tilt-shift lens to an auto-focus camera, the lens is still a manual focus lens and will not auto-focus. All controls are manual (except the aperture which is controlled in-camera).
Above: Another shot with a tilt-shift lens- compare to image below which was shot with a normal 50mm lens using the freelensing technique.
Below: Notice a similar area of focus and tilt as the image above using the tilt-shift lens, but the "sharp" areas are not quite as sharp or of the same quality.
Option: Freelensing
Freelensing is a technique where the lens is detached from the camera body, hand-held, and tilted. You must move the lens around to focus. While it's a plus to use a lens you already own, and while you can create some neat focus effects, there are still multiple down sides to shooing this way. For one, it can be difficult hand-holding both the camera and the lens while trying to focus. They can become heavy, take longer to focus, and even allows for a greater risk of dropping one or the other. The main down side to using this technique however, is the risk of introducing dust onto your cameras sensor. Since the lens is not attached to the camera body, and a digital sensor is charged when the camera is on and operating, it tends to suck dust towards it if there's nothing there to protect it from the elements (such as a lens or body cap).
A few other things to keep in mind if you decide to try this technique are light leaks, exposure, and aperture controls. You will want to use a lens that has an aperture ring that you can manually control, or a lens with a wide-open aperture when the lens is off of the camera. This means Nikon G lenses and Canon EF-S lenses are not good candidates. Since the lens is not communicating with the camera, it's best to predetermine and set your exposure (shutter speed and ISO in this case) on the manual setting. Also because the lens is not attached, the space between the lens and camera body can create light leaks. To prevent this, just be sure to bend the lens towards the light source so that the open space is away from it.
Another freelensing example but with a much more dramatic effect and area of focus
Option: Digital manipulation
Another option in achieving a similar effect is to digitally manipulate your images in Photoshop or another photo editing program. There is a variety of tools and techniques that can be used, such as the blur tool.
Image above is the same exact image as the first one posted at the top (normal with shallow DOF) but with blurred areas created in Photoshop
Option: Lensbaby
One of the multiple Lensbabies available
A Lensbaby SLR lens is an affordable alternative to a tilt-shift lens (which can be pricey). They come in different camera mounts and can fit both 35mm and digital bodies. The effect is somewhat different because the lens has a circular field of focus instead of a planar field of focus like that of a tilt-shift lens. These lenses are known for their "sweet spot of focus", and are also manual lenses, but even more so than the tilt-shift ones. The focus and movement are done by hand, and the apertures must also be changed by hand. There is no aperture ring on these lenses, so they come with aperture disks with pre-cut holes which must be swapped out in order to change the aperture.The quality of these lenses is nowhere near a tilt-shift lens, and is sometimes compared with toy or plastic camera lenses.
Image above taken with a Lensbaby lens
To read more about the topics covered above, or to see some interesting images using these techniques, refer to the links below: 
* The (new) Lensbaby Line- a review on the KEH Blog from last year

* 101 Photos Taken With the Lens Detached

* In-stock items at KEH.com: 
- Lensbabies
- Tilt-shift lenses
- View Cameras