Can they steel your domain name?

Tweet

I did a search on the internet for a domain name for a business. First I did a search for a service like mine to see if it was on the net. Then I did a search for the name and it said there is nothing by that name. So I typed the name into one of those domain search sites to see if the name was available at .com and it was. I didn't sign up right away because I am nervous to pick a spot in case they aren't trust worthy. A week later I did the search for the domain name again at .com and it said it was taken. I have done a search on the net for this but there is no website by that name. Can they steal your domain name if you type it isto a search engine to see if it is available?