he advised me that what I saw was probably malware, and not to be too concerned.
"He" is an idiot. He says malware like it's some sort of specific threat. Malware is short for malicious software, which is a catch-all for
any threat; virus, worm, Trojan, etc.
This one actually showed a window with a list of viruses including 2 Trojan Horses.
Did that screen look anything like the one show here?
http://www.im-infected.com/rogue/user-protection.html
If so, the so called AV software itself is the malware. It's been around for years. Every year they come out with a new one with the current year in the name to try and make it sound legitimate. The list of viruses it "finds" are usually a mix of .dll files, some real, some made up. All Windows stuff since .dll files don't exist in OS X.
On a Windows system, it is very pervasive. Blocking access to the Internet and even deleting your real AV software from the hard drive to keep you from removing it. All the while trying to extort $29.99 from you to activate the "full version" to remove the "viruses" from your computer. If you fall for it, all it does is return control of your computer to you while actually doing nothing (though it claims to have now cleaned your computer). Better than that, the crooks now also have your credit card number.
Anyway, I mention this Windows threat because it's been reported that the same jerks are trying to figure out a way to get it to work against Mac users. Since it can't install without your help as it does in Windows, infected web sites instead pop it up in a window. There is no software anywhere which can scan your hard drive remotely for viruses through a web browser, so it is of course completely bogus.
Wherever you click on the window it would download an exc. file.
I imagine you meant .exe file. Yes, that's one way the infection on a Windows computer starts. Once you download the .exe file, it automatically runs and then downloads the rest of the components it needs from the Internet to take control of your computer. Since an .exe file cannot run in OS X, you just end up with a few files taking up a bit of space on your hard drive.
He told me that Apple had a free virus software that I could use, but I couldn't find it.
Apple provides no AV software. Just links on some of their pages to third party vendors that do.