Among the Sony lenses there are some types with a DT abbreviation and some are without it. Those having DT called DT-series.
What does DT mean? What is the difference between DT-series lenses and the rest of Sony?
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DT (Digital Technology - source) means that the lens is designed to work with a cropped sensor camera. It should not be used with a full frame sensor or 35mm film. If you use it with a full frame sensor then some of the image would be black, particularly towards the corners, as the light projected by the lens would not cover the sensor.
Having a smaller area to project light on to means that lenses can be smaller, lighter and are generally cheaper. However if you later upgraded your DSLR body to one with a full frame sensor you would have to replace the lens.
TL;DR answer: DT lenses have a smaller image circle designed to match APS-C sensors, which are smaller than full-frame sensors. This makes it possible to design high-quality lenses that are smaller, lighter, and less expensive than an equivalent full-frame lens. However, they will not be able to take full advantage of a full-frame sensor.
A DT lens is designed so that the image circle projected by the lens covers the 16x24mm APS-C area, but not necessarily the 24x36mm area required by a 35mm full-frame camera. In the case of zoom lenses, the image circle may cover full frame at certain focal lengths, but this is not true for all zooms. DT prime lenses either do not cover full frame at all, or would exhibit serious optical compromises such as extreme softness in the frame corners on a full-frame DSLR.
When the image circle only needs to cover the APS-C area, several advantages can be realized. It makes it much easier to design lenses with shorter focal lengths, allowing lenses that are truly wide-angle on APS-C. An 18mm lens is expensive and difficult to design and manufacture if it needs to cover full frame, but can easily be done if it only needs to cover APS-C. The smaller image circle allows for smaller and therefore lighter lens designs, especially for lenses with focal lengths up to medium telephoto. It enables superzoom lenses like those made by Tamron to be compact and lightweight while retaining reasonable image quality. It allows fast lenses to be made more easily as well. It also makes the lens significantly less expensive to manufacture. These advantages hold for all manufacturers of APS-C cameras and lenses, not just Sony. In fact, Canon exploits the fact that APS-C camera mirrors are smaller with the EF-S mount, an extension of the EF mount that allows the rear element to be placed farther back than in EF lenses, making it even easier to design ultra-wide lenses (such as the remarkable EF-S 10-22mm lens).
Of course, it means that you won't be able to take advantage of the full-frame sensor in high-end models. On Canon, an EF-S lens physically cannot be mounted on full-frame or APS-H bodies with an EF mount and can only be used on APS-C bodies with an EF-S mount (which can take full-frame EF lenses as well); the rear element may extend past the lens mount far enough to collide with a full-frame or APS-H mirror if an EF-S lens could be mounted on such a body. On Sony, the image area will be limited to that of the APS-C crop area of the sensor leading to a loss of resolution, and exposure metering accuracy may be compromised, because darkened or black frame corners (vignetting) can confuse the light meter and cause overexposure.
According to Sony (you'll see this on all Sony DT lens product pages):
Advisory when using DT lenses with full-frame (35mm equivalent) DSLR cameras
Sony α (alpha) lenses with the Digital Technology (DT) designation are ideal for use on APS-C size DSLR cameras and are not recommended for use with full frame DSLR camera bodies (such as the DSLR-A900). Although DT lenses can be used with full frame cameras, they can only realize a maximum of 11MP image size (when using the DSLR-A900), and are not guaranteed to provide accurate Auto Exposure performance.