Can they charge an account without authorization?

So, today I recieved a call from allied interstate. When they mispronounced my name, I said that I wasn't there and offered to take a message. I took down the phone number and asked what the call was regarding, to which they replied it was highly personal. So, 20 minutes later (after browsing this site and realizing they were probably going to try to mess with me), I called them back, prepared to deal with whatever they were going to try to throw at me (I have had poor credit in the past, and even taken out a loan to consolidate my credit card dept - thank god I finally got it all under control). Of course, they wouldn't give me the guy who originally called and they asked for my phone number and zip code. I replied that they had called me and that I wasn't comfortable giving out that information. Luckily, I called from the same line as they had called me from, because I had considered calling on my cell phone on my way out, but decided not to since they may just trace the call... which they did. They asked me if I was who they were trying to reach and I told them that yes, they had the right name. Then they told me I had an account open with American Compass for $115... so I ask, who in god's name is American Compass? He tells me that they are a mail-order book company, to which I reply that I had never used and had never received books from. Apparently, he took my word for it (but I guess we'll only know if I get a call back) and said they would dispute the claim and "take care of it". I asked exactly what that meant and if they had access to my credit cards to just pay off the amount without me authorizing it, to which he replied, "That would never happen." Is he lying? Obviously, this company is a joke and they're just out to screw people, so should I still be worried? And what do I do if they do charge something without my permission?