The ability for a touch surface to respond to multiple finger touches at the same time.
For example, a common use for multi-touch is pinch-to-zoom, where you can place two fingers on a screen image (like photo, web page, or map) and spread your fingers to zoom in, or pinch your fingers together together to zoom out.
Another example is tapping (or dragging) with two or more fingers, and having that do something different than only using one finger.
The multi-touch sensing doesn't work on the resistive type of touchscreens (budget touch phones), the capability comes inherited in capacitive touchscreens (majority of touch phones nowadays).
Apple is considered the pioneer of multi-touch technology on mobile phones with their first iPhone. They introduced the pinch-to-zoom and rotate gestures that are now available across various platforms.