They are a scam.I think. the phone rings, and there is this guy with a heavy accent on it, and he says he is from "techincal support" and he wants me to open my computer and type in this address at the top that says "ammyy.com" and I do that and this box comes up asking for a password of some sort.

I got suspicious and stopped there and began to question him as to who he was with, why he was calling, etc. and his answers were vague and sounded not on the level, so I asked for a supervisor and this other heavily accented guy got on and also did this runaround bit was a bit rude, so I asked him his company name, his phone number and his website, which he gave me. It was www.ostechadvisor.webs.com. He never answered my questions to my satisfaction as to why he had called, etc. so I told him I would do some research and call him back if I thought I needed his services and hung up. He also said he was calling from India.
I found that is a free website and they are not legitimate.
The only reason I went as far as I did with him in the first place was that I had trouble connecting to the internet the other day and my service provider people had just been out on a service call and got it working again a few days ago. So, did I do the right thing? What do you think these people were up to? Did just opening that box like I did before I terminated cooperating with them put me at any risk?

Am I an idiot?

They are a scam.I think. the phone rings, and there is this guy with a heavy accent on it, and he says he is from "techincal support" and he wants me to open my computer and type in this address at the top that says "ammyy.com" and I do that and this box comes up asking for a password of some sort.

I got suspicious and stopped there and began to question him as to who he was with, why he was calling, etc. and his answers were vague and sounded not on the level, so I asked for a supervisor and this other heavily accented guy got on and also did this runaround bit was a bit rude, so I asked him his company name, his phone number and his website, which he gave me. It was ostechadvisor.webs.com . He never answered my questions to my satisfaction as to why he had called, etc. so I told him I would do some research and call him back if I thought I needed his services and hung up. He also said he was calling from India.
I found that is a free website and they are not legitimate.
The only reason I went as far as I did with him in the first place was that I had trouble connecting to the internet the other day and my service provider people had just been out on a service call and got it working again a few days ago. So, did I do the right thing? What do you think these people were up to? Did just opening that box like I did before I terminated cooperating with them put me at any risk?

Am I an idiot?

No,  Not at all an idiot. to many people use the word 'idiot' to describe anyone with whom  they don't agree. Let's just say that you need to become more wary of Strangers who unexpectedly call for questionable reasons.  Especially if they sound like some Billion dollar retail store's  cusomer service department.

correction:    should read   Too many people

Jerome,  I don't think that opening the address is going to do anything.  If you had entered a password and followed the instructions that they were about to give,  you would have turned over control of your computer to them  remotely.  Then you would have seen them scanning and moving all around on your screen and files opening up before your very eyes.  You should never click on any unknown "link" given by strangers as in many cases it is to feed your computer a trojan virus.  It can snoop out information such as passwords,  bank account numbers etc.  I  think you are ok since you terminated it and in the future,  if anyone calls and gives you the same speel,  just say no thanks and hang up because it is definately going to be a scam.   good luck