I was terminated from my security job because I let an unauthorized person into the building; but I didn't know he was not supposed to be in the building. He came into the building with proper ID but his access on his card was disabled. Because of this reason the company let me go. My question is if I go apply for another security job what do I put on the application for reason for leaving?
I would not say "dismisssed" or anything negative -- the new company will likely imagine the worst, and may not even call you back.
Say something simple, general, and neutral. You're probably feeling bad/guilty right now and in a confessing mood, but that will work against you. I've been fired twice, and got new jobs by explaining it simply, and not dwelling on the past -- most companies are far more interested in your experience and what you can do for them, as opposed to what happened with your previous employer.
People make mistakes. Do not say anything about it unless asked, and if asked, explain it as you do here, indicating that you have learned from your mistake and will be extra careful in the future. Do not blame your old company -- on the contrary, say you understand completely their actions.
Use your mistake to your advantage as much as possible. Convince your new company that you never want to make a mistake like that again.
Dismissed for breaching security is about the worst thing next to stealing in that industry. Best not to mention it at all. Expect to be asked why you left your last position and have that impact badly on your application. There is no sense lying about it, it's too easy to find out unless you leave the locale.
Personally I would change industries. Security skills can be used elsewhere (I'm assuming you're licensed and trained). Also security jobs encompass a huge range of activities, you might want to look into a change that way.