Anthony,
I have had firsthand experience with this (acting on behalf of the FAA) multiple times in the past few years. I do not fully agree with Zeldman's description of what is likely to happen from here. Not saying it's impossible, but in my experience, highly unlikely. Especially here in the northeast, PDs, even those involving penetrating a POTUS TFR, don't tend to immediately result in Enforcement Actions.
The sequence of events will be something like this:
The PD is written up and the FSDO with jurisdiction will assign the case to an ASI.
The ASI will have a small stack of similar PDs and will go through them one by one. In my experience you're looking at anywhere from 2-4 weeks before you get a call, but that's not set in stone.
The ASI will make initial contact with you. It is my strong recommendation that you "demonstrate a compliant attitude," withhold nothing and admit to everything. I realize this is contrary to the advice of some attorneys, however, in the recently renamed "Compliance Program" (former the "Compliance Philosophy") it is a critical component in the decision tree of how the FSDO will handle the deviation. He will likely ask you to submit a brief written account of what happened, either by email or some other means. I would recommend that you describe where you fell short of your pre-flight planning obligations (14 CFR 91.103), how you'd better prepare for a flight next time, and so on.
After they've spoken with you and received your account of the PD, they decide what to do with you. There are multiple options: Enforcement Action (rarely used these days except for egregious violations) or Administrative Action. (There are some sub variants but they don't matter for this scenario.)
Worst case scenario first: once they decide to go with Enforcement Action, it's over. You will have a proposed certificate action sent to you, and almost nothing will stop the chain of events from unfolding to the point that you're dealing with a suspension of airman privileges. It's possible that with a properly submitted ASRS form you will not have to actually serve the suspension, although it would still go on your record.
However, the good news is that's extremely unlikely, especially if you followed my guidance above. Most likely it will be an Administrative Action, and most commonly they'll assign RT (Remedial Training) with a local CFI, often a FAASTeam rep. This is how I know a fair amount about this subject - I've conducted numerous RTs, the majority of which have been for TFR penetrations, including VIP TFRs. We get a lot of presidential visits up here in the summer, and we've had upwards of 32+ reported incursions per visit. (Ugh... )
If RT is proposed, and you accept (and in my humble opinion, you'd be crazy not to), you have to agree to it in writing. It becomes a binding contract between you and the FAA. In many cases, by this point the ASI will have turned the actual administration of the RT over to another ASI who is frequently the FPM (FAASTeam Program Manager.) Due to the FPM's connections with the GA community he or she is often best suited to assign the RT to an experienced local CFI. You must agree to bear all costs for the training, including paying the CFI his/her hourly rate.
The syllabus typically includes a mix of online training from FAASafety.gov (typically videos and/or interactive CBT of some kind.) Then there will be a specific syllabus of training, usually just ground training for TFR incursions, but it is possible there would be both ground and flight training required. It's usually anywhere from 6-10 hours of total training, including both the online portion as well as ground/flight.
When I conduct RTs I try to attend a meeting at the FSDO with the ASI and the airman as a general "pre-brief" and to put the subject of the RT at ease. Once RT is assigned, the hard part is over. You just have to do your part, demonstrate a compliant attitude throughout and complete the training on the appropriate timeline. You will be given a deadline -- take it seriously. The ASI may or may not extend the cutoff date due to personal reasons. By way of example, we had one RT in which the airman had a vacation planned overseas which would have made the proposed timeline for the training syllabus impossible. The ASI's attitude was that it wasn't a good enough reason to delay the Remedial Training. The airman's options were to give up the RT option or cancel his vacation. He chose the latter.
In my experience the only way this doesn't go the way of RT is if 1) you're combative on the phone with the ASI, 2) avoid phone calls or letters on this matter from the FSDO, or 3) you penetrated the inner core of the VIP TFR. I won't make the call on that last one based on your limited description of events. When you poke a hole inside the 10nm ring the odds of Administrative Action diminish rather sharply.
If those three things don't become issues, in my experience you have a better than 95% chance of going the RT route. Remember... admit to everything, conceal nothing, don't make excuses. That first phone call is pretty important.
Best of luck,