These are similar varieties of finger pressure massage. They are both based on applying pressure to a pattern of specific points that correspond with the acupuncture points. Pressure is applied with the thumb, finger, and palm.
The goal is the efficient and balanced flow of chi through the meridians. It is believed that where there is tension being held in the musculature, the flow of chi is impaired through those areas, which can lead to chronic problems not only in the musculature but in the associated organs. Stretching and movement are also sometimes used.
Acupressure is the more generic term used for this approach and shiatsu is the Japanese version.