The most important are the ones that are needed to help the decision at hand. This will depends on the context and who looking at images.
- Date & Time - Are probably most important. They are often different than the file-data since that may reflect when the files were copied, shared, sent, edited, etc. To look for a photo of an event or to find the time-of-year when things looked like that, this is key.
- ISO - Along with resolution (Metadata but not part of EXIF) determines the maximum possible print size for an image.
For the photographer himself and to learn from other's work:
- Aperture & Shutter-Speed - This is the basic of exposure and that lets you know how a shot was taken. If you are the client looking for a photo, you really do not care but another photographer may.
- Metering mode & Exposure-Compensation - Same as above, good for photographers to know how a shot was taken. Even the one who took the picture can use that information to learn and improve his technique.
- Focal-Length - In 35mm-Equivalent terms, otherwise it becomes very difficult to compare. This is important to manage photo gear, decide which lenses to take and which ones to leave behind.
- Lens Model - Also helps with trip preparation or purchase decisions for others.
- Camera Model - Various uses, sometimes to look for specific quality or image characteristics.
For client requests:
- Keywords - The first things clients look for is a photo of a specific subject or concept.
- Resolution - Clients have a print size in mind and resolution together with ISO is required to determine which photo is usable.
- Aspect ratio - Some usage like publications, advertising, banners, etc require a specific aspect ratio. Note that the final aspect-ratio is more important than the shot aspect ratio which is mostly fixed by camera (contrarily to consumer models, only a few pro cameras support multiple aspect ratios).
There plenty of other useful ones and sometimes we use certain data to compensate for inadequacies in others, particularly keywords. For example, it is impossible to keyword all images for every possible future use. Someone may ask for a photo of something which is not keyworded but I known that I probably took it with such lens or camera.