AP (no DHCP server) serving IP's while main router (DHCP server) is not available?

Well, since you edited the exact sentence, and it still implies that there's a DHCP server:

No matter how you word it, it would help if you explained htis.

Why won't you just tell us:

"when I connect a device, it gets an address, and the DHCP server has x IP and y MAC"