What is conservatorship in child custody?

My kids’s dad and I have separated and our 3 kids have been going back and forth between us. We have no court order for custody. The father has now filed for ‘conservatorship’ of our children and wants them to live with him full time. He is also asking that I be ordered to pay child support. What does conservatorship mean in terms of custody and how do I respond to his court paperwork? Do I need a lawyer to respond? Can he actually get child support from me if he makes significantly more than I do?