APEs are used to address non-trivial enhancements that require much discussion and thought. This is intended to mirror the long-standing Python Enhancement Proposal process, but generally not quite as formally. Normally a proposal goes through various phases of consideration. Normally discussion is expected to take place using existing mechanisms (astropy-dev, github, hangouts, etc), and eventually a decision is made regarding whether the proposal should be accepted, rejected, or modified.
It is important to note is that there is not much point to making proposals unless someone or some group has signed up to implement it if it is accepted (normally this would involve the author or authors of the APE). Just issuing an APE in order to spur others to do work is not generally going to be received well. Generally, an implementation is expected before an APE can be considered fully accepted. For proposals that require extensive work that few are willing to perform without some assurance it will be accepted, we can issue provisional acceptance. For serious consideration it is usually good to show that detailed technical aspects have been played with in real code rather (even if it isn't a complete implementation).
New APEs should be created using the APEtemplate.rst file in this repository. Just fork the repository, copy APEtemplate.rst to APE#.rst and issue a PR with that file once you've written it up. Be sure to look through the PRs in this repo first so that you choose an APE# that reflects both those that are in the repository and those that are under discussion.