The things that the leaves originate from are called pseudobulbs.
Wickedthistle I would cut off the parts that have dried up. It might make a few side branches from the healthy part if it is not broken way down at the base. Don't be surprised though if it only makes a few flowers if that. Oncidium are not big rebloomers off old inflorescence. Your next big show will come when the newest growth starts maturing and makes a new inflorescence.