What is an Air Defense Identification Zone (ADIZ)?
What is an ADIZ?
FAR 99.3 says:

Air defense identification zone (ADIZ) means an area of airspace over land or water in which the ready identification, location, and control of all aircraft (except for Department of Defense and law enforcement aircraft) is required in the interest of national security.

To put it simply, if you're going to fly in the United States ADIZ, the United States government wants to know:

Who you are
Where you are
What you're doing


What do I need to have before I fly through one?
A few things, actually.  They're listed in AIM 5-6-1

You generally need to file a flight plan (but there are exceptions). (FAR 99.11)
You need a two-way radio. (FAR 99.9(a))
You need to have a Mode C transponder, and it has to be on. (FAR 99.13)
If you're going DVFR (rather than IFR), you need to give an estimated time of ADIZ penetration at least 15 minutes before entering it. (FAR 99.15, also reference AIM 5-6-1.4b)


What do I need to do while I'm in one?

Remain no more than 5 minutes off the proposed ETA to your various waypoints
Remain no more than:

10 miles from the centerline of your route, if over land
20 miles from the centerline of your route, if over water