What is a Tilt-Shift Lens?
If you remove the back cover of your SLR lenses, you'll realize they all transmit the light to the sensor through a circle-shaped piece of glass. The front element has indeed the same shape, yielding the barrel form of the lens. Full format sensors, like Nikon FX, barely fit into this circle for maximizing glass usage and coping with the mismatch between the lens (circular) and sensor (rectangular) shapes. With the smaller APS-C sensor, like Nikon DX, this fit is more generous. This is why DX sensors perform so well with FX lenses - the corner areas, where image quality degrades due to vignetting and lower sharpness, are excluded from the final image. The unique feature of tilt-shift lenses is their larger image circle, allowing for changing the position of the sensor plane (shift) and the orientation of the focus plane (tilt) within this circle.
Top Features when Doing Landscape Photography
Exceptional image quality. With no shift or tilt applied, you get a winner prime lens - Soft/dark corners excluded! And even more with DX sensors, which, however, also have some pitfalls when compared to FX (see this nice article in photographylife).
Extremely high resolution panoramas (shift). Just shift the lens in the desired direction and stitch the 3 resulting images for an increase of ca. 70 % in resolution. With 24 MP (D3X) and 36 MP (D810) Nikon bodies, the final output is a whopping 41 and 61 MP, respectively. A tilt-shift lens can thus convert your SLR into a medium-format camera for amazing prints at crazy high DPI resolutions (see the "ultraprint" concept by Ming Thein). If shifting horizontally, and depending on how you orient the camera, both extra-wide 3:1 panoramas (horizontal) and more classic 4:3 compositions (vertical) can be created. Super-tall images can also be created by vertical shift (raise), especially by setting the camera in the same orientation. A peculiar feature of the DX format on tilt-shift lenses is the fact that, being smaller than FX, it allows for taking more pictures within the lens image circle. Therefore, at equal sensor resolution, DX will yield a higher resolution photo.
Full control over perspective (shift). If your jungle is the city, architectural landscapes do also benefit a lot from using these lenses. By shifting the lens vertically, the converging lines of tall buildings easily revert to more natural-looking parallel. This feature may also be useful out in the wild, especially if shooting 24 mm or wider.
All-in-focus images at top-quality apertures (tilt). By tilting the lens focus plane to a position parallel to that of the camera sensor plane, both near and far subjects can be put in sharp focus at the optimal f/8 aperture, avoiding the IQ degradation caused by diffraction at apertures narrower than f/16. This is a complex feature requiring practice and mental 3D vision. It can be accomplished by an iterative focus and tilt process. In practice, this is seldom needed since many landscapes resolve well at f/11-16 just by using the hyperfocal distance trick (you may read this page in the luminous landscape). It can still be useful in extreme near&far compositions, when f/16 is not sufficient and/or not good enough in terms of IQ. It can also be used for enabling fast shutter speeds with wide apertures and yet get a deep depth of field.
Full control over focus (tilt). Tilt can also provide highly selective focus for more original and artistic photos. This is for instance the basis of the famous "toy effect".
Combined Effects! All these tilt and shift magics can be used together. One powerful application: A landscape composition is rendered with plenty of detail by using the lens shift and, at the same time, tack-sharp all across the frame by using the tilt feature (see this page at dpreview).
But nothing is perfect...
Heavy and expensive. Nor comfortable neither worth unless you take your hobby or profession very seriously. At least it makes sense: lots of extra glass and high quality metal barrels add weight at a certain cost.
Fully manual lenses. Which means...
Lots of work! Careful picture planning and previsualization, mandatory use of tripod, extra post-processing required, exposure and WB accurate preset, live-view (battery-eater) abuse, long focus-tilt-shift loops...
Be aware of issues resulting in limited use of modern PC-E lenses with some Nikon cameras, mainly due to obstructive flashes in some bodies (see this article by Ken Rockwell).
Nikon PC-E lenses only combine either shift (left-right) with tilt (up-down) or raise (up-down shift) with swing (left-right tilt). This satisfies my needs, but perhaps not yours! If so, this limitation can be fixed by Nikon Service (at a cost). Otherwise, consider Samyang and Canon alternatives.
For a more comprehensive theoretical introduction on the most complex tilt feature, I recommend this excellent article at Cambridge in Colour.
Good luck and enjoy!