"Family" in general means your closest relationship. That varies based on your circumstances. If you're married, it should definitely be your spouse. However, you may not be, and so it will be somebody else whom you're closest to, such as a brother or sister, a parent, a best friend, etc.
A Dream Chaser Peer is a fellow executive director you regularly collaborate with in an official capacity. This does not mean somebody you call up randomly or even regularly to talk to. This is someone who is your partner in collaboration.
This might include writing down a list of questions, reading notes from a previous talk, looking through relevant resources, completing pertinent tasks, etc.