Can you use regular headphones for a Nintendo DSi?

Any pair of headphones with the basic 3.5mm jack will work on the Nintendo DS.

Headphones with microphones (except the ones made for the DS) will be longer and won't fit well. You will still get audio, but it will probably be distorted. The microphone on the headphones won't work.

However, there are special headsets for the DS, which fit into the oddly-shaped rectangle on the headphone jack, for an external microphone to be supported.