There are indeed differences, and the purpose of each is not necessarily the same either. Of the four, three do aim to improve your dynamic range in one way or another, however Multiple Exposures has other uses.

To start with HDR, this is more of a post-process blending technique that merges multiple photos of the same scene to improve total dynamic range. It is most useful when the scene you are trying to photograph has considerably greater dynamic range than the camera is capable of capturing in a single shot. Usually only three shots, one normally exposed, one under exposed by a stop or two, and one over exposed by a stop or two...are required to blend an HDR scene with a tool like Photoshop or PhotoMatix. Ultimately, the final step in HDR processing is to convert a 32-bit floating point HDR image into a normal 16-bit or 8-bit LDR (low dynamic range) image. Currently, there is no known device capable of actually showing the full dynamic range possible with an HDR image, and one must selectively tone map the image to produce a meaningful LDR result. HDR simply gives you maximum control over which tones you keep, which you discard, and which get "compressed" into the lower dynamic range of a normal image.

ADL, or in the case of Canon cameras HTP (Highlight Tone Priority) is a mode of camera operation that automatically tries to preserve as much highlight dynamic range as possible. This usually results in under-exposure in one way or another. If highlights are your primary concern, this mode can help ensure that you don't fully blow out your highlights such that they are unrecoverable. Unlike film, digital sensors have a hard limit on their sensitivity, and once that limit is reached, no further detail can be captured, and any additional exposure will simply bleed out detail information that you have captured. The consequence of using automatic camera modes that try to preserve highlight tones is that you might end up with underexposed shadow tones. This may or may not be an issue in the general sense...if highlights are what your after in your shot.

Automatic Bracketing is a feature of many modern digital cameras that aims to automatically adjust exposure for bracketing shots. As mentioned before, bracketing is usually a necessary starting point for HDR processing. Multiple shots, usually three, are necessary to blend an HDR image. Auto exposure bracketing allows you to set how many stops you wish to bracket, up to the standard metering range your camera is capable of (usually -2 to +2 EV). When you actually take a shot, you press the shutter three times in a row to capture normal, under-, and over-exposed shots.

Multiple Exposure photography is a bit more versatile. It is most effective with film, although there are techniques that you can use to achieve a similar effect with digital...its just more post-processing work. The general idea with multiple exposures is that you don't advance the frame...one exposure is taken on the same frame of film as the previous. In the case of a lunar eclipse, this will continually increase the exposure of the sky while also capturing the moon in multiple locations over the total duration of all exposures. The effect is a compound of both a elongated night sky photo, as well as the composite of numerous individual shots of the moon as it progresses through its eclipse. One need not expose the same scene with multiple exposures, however. Exposing multiple scenes with a single film frame can produce a variety of interesting effects, only really limited by your imagination.