The best time to divide is after blooming, before the plant starts its next growth cycle. Division can take place at other times but may result in slow or stunted growth for a cycle and/or skipping a bloom cycle.
When you take the cutting, take one from a "lead" which is where new growth will come and make sure you take at least 3 mature pseudobulbs. Less will result in the division being set back.
It looks like an Ondicium of some kind. Go to the AOS website (
www.aos.org) and get a free culture sheet on Oncidium care. Or I suppose you could just grow it outside just like your neighboor grows it. It clearly loves those conditions.