OK, I did a little digging and here's what I can definitively tell you:
1-866-GA-SECURE is a joint venture between AOPA's "Airport Watch" program and the TSA.
The hotline is staffed by federal officials and can be used for reporting suspicious or unauthorized activity - e.g. the theft of an aircraft, someone prowling around on the ramp after hours, etc.
The folks answering the hotline will be able to contact local law enforcement (and being that they're federal officers making the call instead of Joe Pilot, they may get a quicker response than you would), but police dispatch isn't their primary function.
The TSA's own guidance on this hotline indicates that it shouldn't be your first resort -- they would prefer that you contact:
Particularly at fields that are attended 24/7, and those with security or police presence, calling the folks on the field will often give the fastest response, and they should be your first call.
The TSA Hotline (1-866-GA-SECURE)
For fields which are not attended / don't have a security or police presence, this can be an effective way of getting law enforcement to the field quickly.
911 or other local law enforcement emergency numbers
In situations with immediate and potentially dangerous consequences, or ongoing criminal activity (breaking into aircraft, bypassing gate locks, etc.) calling local law enforcement's emergency number / 911 is The Right Thing To Do.
In all cases you can follow up with a call to the TSA hotline to confirm they're aware of the incident. Whether on an initial call with an ongoing situation or a follow-up the hotline folks will want the basic 911 information (Who you are, Where you are, and What your emergency/situation is). Obviously depending on the situation they may ask for additional details - e.g. if you're reporting that a plane was stolen it would be very helpful if you could provide the registration number.
AOPA's GA Security course covers a bit more of what you may be asked for / about, and the kind of things you should be looking for when informing any of the above authorities of suspicious activity.
Who responds to your hotline call and how quickly they respond is less definitive: It will vary depending on where you are and the local resources available. A grass strip many miles away from the closest police presence at 3 AM will obviously have a longer response time than a 24x7 attended field with a police presence on the field, no matter who you call.
Typically the first response will be airport personnel (if the field is attended) or local law enforcement (and obviously that will be the case if you're calling the airport's management/operations office or the local police directly). Whether or not federal authorities get involved in the response would likely depend on what you're reporting.