Dithering in astrophotography is the slight offsetting of stars in each and every sub frame (sub). Dithering these days is usually achieved by pairing two compatible programs, such as PHD2 (guiding software) and BackyardEOS (image sequencing software), or SGP and either PHD2 or MetaGuide, and instructing the image sequencer to dither. During imaging, between each sub, or between a given configurable number of subs, a command will be sent to the guiding software to dither.
Dithering is generally achieved by slewing the mount slightly, then allowing the guide program to return the star centroid to proper guide position. That results in a shift in the star position of each sub. Depending on what kind of imaging equipment you are using, you may need to use more aggressive dithering settings. Monochrome CCD cameras can usually get away with much lower dithering, so a lower aggression setting that offsets stars by a couple of pixels is usually fine. DSLR cameras, which are often not optimized for astrophotography, and often have higher levels of pattern noise than CCD cameras, usually require more aggressive dithering. Stars should be offset by many pixels between each and every sub frame.
The purpose of dithering is to offset stars relative to pattern noise in each frame. Pattern noise most often presents as hot and cold pixels, but may also show up as bias pattern (usually vertical banding) or other pattern (fixed or non-fixed) which can include horizontal and vertical banding, salt and pepper noise, etc. Offsetting the stars, which are your alignment foci, relative to the frame allows Sigma Clipping integration algorithms to identify and reject sigma outliers. Many stacking programs support sigma rejection, including DeepSkyStacker (DSS) and PixInsight's Batch PreProcessing script (BPP). For dithering to be effective, you usually require a minimum number of subs for the rejection algorithm to actually function properly. The minimum recommended sub count is 10, however more will improve rejection and result in cleaner frames.
As an example of using and not using dithering, here are two of my most recent deep sky images. The first, Orion's Sword, was dithered, while the latter, Horsehead and Flame nebulas, was not dithered (by accident, simply forgot to enable dithering). The latter image has more noise, despite being imaged at the same dark site with the same general settings.
A more apt example of the consequences of NOT dithering can be seen in my recent Comet Lovejoy image, where I used very short, undithered exposures for the comet:
The remnants of heavy banding, due to the shorter exposures, can clearly be seen in the full size image. I could have dithered (probably should have), which would have dealt with a lot of the banding and hot pixels. I had extremely limited time to image the comet (on the one unexpected clear night for weeks) before the moon rose, so I went for the fastest imaging I could. If I have another opportunity, I'll be dithering for sure.
It should be noted that dithering, while often used as an alternative to taking dark frames, is not necessarily a replacement for dark frames. Dark frames contain all of the undesired dark sensor signal. That includes things like amplifier glow or other glows. If you have issues with amplifier glow, then you will still want to take darks. This can be difficult with DSLRs, which lack any kind of thermal regulation. With CCDs, it is easy enough to take reusable sets of dark frames for various temperatures (i.e. a winter set, spring/fall set, and summer set are common with CCD.)
Many image sequencing packages support dithering. BackyardEOS (and BackyardNikon, although still in beta) is an excellent beginner package for those using DSLRs for imaging. They are fairly simple and easy to use, and have compatibility with PHD2 (Push Here Dummy). A similar program is APT, or Astro Photography Tool. APT is more feature rich than the Backyard* programs, but generally does the same thing. Both applications are in the same general cost class,
PHD2 and Metaguide are the two key guiding programs people usually use. PHD2 has broader compatibility and is definitely easier to use. Metaguide (MG) compatibility is sparser, and it can be more complicated...however it employs lucky imaging techniques to produce much, much more accurate star profiles and star centroid calculations, and can effectively guide at a much faster rate, producing more consistent guiding and tighter results. Sequence Generator Pro (SGP) is another sequencing package that is compatible with both PHD2 and MG. It is a more expensive package than BackyardEOS, much more complex, and is generally designed for CCD imagers. It is extremely powerful if you have need of it's features, which support fully automated imaging setups.
Finally, there is MaxIm DL. MaxIm is an older, well established fully featured package for astro imaging. It does everything, including image sequencing, guiding, image processing, etc. It is one of the most expensive packages around, but if you need the most advanced features and fully automated software, this is probably the one to get. It also has support for PHD2 and MG guiding, if you prefer.
When it comes to PHD2 guiding, the most common. The best configuration for the most effective guiding and dithering is to connect to the mount directly via ASCOM. This is in contrast to connecting to the mount via the ST-4 cable that connects to the guide camera. There are limitations when using ST-4. PHD2 can either connect to the mount "On Camera", or you can find and select either the ASCOM Chooser or directly choose your mount (if it is listed). Either of those options will usually support more effective guiding and more accurate dithering, so it is highly recommended.