Why Can't I Download .dmg files????

Up until I started having attitude problems from Firfeox 24, my computer was running quite well. I got fed up with Firefox 24 and decided to try a different browser. I tried GoogleChrome, as well as Safari. I also had MacKeeper downloaded on my computer for a short time, but, after seeing warnings about it in the forums here, I chucked it! So, With Safari and GoogleChrome, I discovered a strange problem-I can't download .dmg files. I can't update Adobe Flash Player, I can't download files for Apps..it'll look like it is starting to download, then it will stop and say "Plug-in Failure". I've been using the Opera Browser that I downloaded for free from the App store......that so great! I'd like to be able to download these certain files again and attempt using Firefox 25. I am on a late 2006 Intel-Based iMac Desktop computer which is running on Snow Leopard. If someone can help, I'd be most grateful. Thank you!