Apple warned that it would eventually want notarization for non-App Store software running on macOS Catalina, and now there's a firm deadline for that request. The tech firm has informed developers that all of these apps will have to be notarized from February 3rd, 2020 onward if they're going to run "by default." Older apps will still run trouble-free, and you can still run non-notarized apps if you're willing to jump through a few hoops -- this will just prevent you from launching newer apps from beyond the Mac App Store without doing something else first.
The company has required notarization for apps (which gives them a Developer ID) since macOS Mojave 10.14.5, but warnings will soon become error messages for Catalina users. Apple had originally intended to clamp down when Catalina was available, but decided on a transition period to help developers adapt.
As you might imagine, this is all part of a push for greater security. The notarization not only prevents people from 'casually' running malware, but lets Apple take action if there's a major security flaw or another serious problem. You'll have to understand the risks if you want to run software without those safeguards.