A Nuzlocke is a challange in Pokemon games. The rules are:
1) You can only catch the first Pokemon on each route/area. If you kill it/it runs away: tough luck.
2) If a Pokemon faints, release it.
3) Nickname all Pokemon.
You can also add more rules like one pokeball per town, no healing items in and/or outside of battle, only use a Pokemon center before a gym battle, etc....