Ok. Now the real answer.
As others have said, if the case was dealt with via Accelerated Rehabilitation (AR) or something like that, you are clean. As long as you complete the terms of the AR, your record is wiped clean.
Also, page 3 of DPS-799, the pistol permit form, it explicitly tells that you do NOT have to disclose arrests, only convictions.
http://www.ct.gov/bfpe/lib/bfpe/bfpedeclrule.pdf
Now lets assume worst case. You were convicted of a drug crime.
What was it for?
How much did you have?
Why is that relevant? Because that is what the law is.
From the CT Office of Legislative Research list of 11 offenses that prohibit you from having a CT Permit to Carry Pistols and Revolvers a.k.a. a "Pistol Permit"
11. first offense involving possession of (a) controlled or hallucinogenic substances (other than a narcotic substance or marijuana) or (b) less than four ounces of a cannabis-type substance (CGS § 21a-279(c)).
So if you were busted for Heroin, you're out.
If you had a personal amount of weed on you. No problem. Actually, if you were a dealer and had less than 4 oz of weed, you are fine.
See item number 3:
http://www.ct.gov/bfpe/cwp/view.asp?a=1838&Q=418126&PM=1
Whatever you do, don't lie on the application.
Please people. MA law is not CT law. Also, if you are going to talk about the law, please. PLEASE use citations, so we can read it ourselves. Otherwise your advice is worse than useless. It may well be wrong.
Don
p.s. The BFPE site is FULL of great information. Remember, they are truly, honestly, impartial. It is a citizen panel and they are the ultimate authority on firearms issues in CT. This is what differentiates us from MA, to a large degree. It is because of the BFPE that we are a state where as far as firearms go, the LAW is what matters. Not what your local PD thinks. Familiarize yourself with the BFPE. Go to a meeting. Write your state rep and tell them how much you appreciate it. (so they don't try to put it under the control of the DPS again, like they tried to do last legislative session)
If you have a disqualifying item on your record, your only recourse is to try to get pardoned.
If you have something on your record that is not on the list, but your local PD denies you illegally, you can and should appeal to the BFPE. If you don't have one of the listed offenses, you WILL prevail with the BFPE. Remember that thing I said about the rule of law. In most cases you don't need a lawyer, since nobody is disputing the facts. You and the local PD both agree on all facts of the matter. i.e. you were arrested for A and convicted of B and thats it. The conclusions drawn by each of you are different. But again, the law is the law.
http://www.ct.gov/bfpe/site/default.asp