Runs is just a term for "play through", meaning, a normal play of a game. Like just a regular run through in Pokemon, you just play the game like normal. A Challenge takes a regular play through, and gives it rules to make it a bit more challenging. Whether there are rules such as: using only one color of Pokemon; using only one type; using only one Pokemon.
The most common challenge, or most advertised, at least, is the Nuzlocke! Each Nuzlocke challenge has a different set of rules, except for three rules:
1. You may only catch the first Pokemon you meet on each route. If the Pokemon faints, you may not attempt to catch any other Pokemon.
2. If your Pokemon faints, it is considered "dead" and must be released or boxed permanently until the challenge is completed.
3. You must nickname all of your Pokemon in order to create somewhat of an emotional bond with them, making the challenge more difficult.