Showing allows the reader to ‘see’ the events of the story as they are being played out either through words, action, or “basic objective description of objects or settings that a reader would naturally see” if actually there.*
Telling is a summary or narration of what is happening, backstory, definitions and explanations, and “any analysis of or commentary on what is happening in the story.” *
Traditionally one should show the most important parts of the story, the things people do and say, the way they interact with one another, dialogue and anything else that “changes the situation of the story.” Telling should primarily be used “to fill in the gaps, to supplement what is being shown, or dramatized.”
Most stories are a mix of both. The hard part, of course, is doing both well. Like this:
Lyra and her dæmon moved through the darkening hall, taking care to keep to one side, out of sight of the kitchen. The three great tables that ran the length of the hall were laid already, the silver and the glass catching what little light there was, and the long benches were pulled out ready for the guests. Portraits of former Masters hung high up in the gloom along the walls. Lyra reached the dais and looked back at the open kitchen door, and, seeing no one, stepped up beside the high table. The places here were laid with gold, not silver, and the fourteen seats were not oak benches but mahogany chairs with velvet cushions.
* Alice LaPlante/The Making of a Story