I've seen many people saying how some sites can infect your computer without actually clicking anything. They said that it was a "Flash Drive By" and it would use flash to download a file of any kind. Does that really exist or could it be something else?
It's not called Flash Drive By, but Drive-By Download, and yes, it's basically downloading malware just by visiting an infected website. Usually drive-by downloads work by exploiting a browser vulnerability (or a vulnerability in plugin like Flash or Adobe Reader), which leads to remote code execution triggering the download of malware.
You can learn more about Drive-By-Downloads, the mechanism of infections, the statistics etc. in the following OWASP presentation.
Yes, it exists. There have been a lot of vulnerabilities in Flash that allow code execution if the user views a flash file (which most browsers will do automatically, if you have the plugin installed).