Flash has been a high-value target for exploit developers for years, particularly because of its near-ubiquitous installation base and the fact that (historically) it will generally run automatically whenever a page with Flash content is loaded. This makes it very easy for a large number of systems to be targeted and compromised with a single exploit.
As it turns out, Flash apparently has a lot of flaws for those developers to discover as well. This has resulted in dozens upon dozens of Flash-related CVE entries every year, with many of them starting off as zero-day exploits. Flash Player is currently #16 on CVE Details' Top 50 most vulnerable products list, with a total of 510 vulnerabilities discovered since 2005. The last year in which there were less than 50 vulnerabilities discovered for Flash Player was 2009, and there's already been 132 discovered this year!
Flash Player Vulnerability Statistics
The vulnerabilities that have been cropping up aren't trivial things, either. Nearly 84% of Flash Player vulnerabilities discovered to date are usable for code execution, and about 80% of all Flash Player vulnerabilities are rated with a CVSS score of 9 or higher!
So, what can happen if you keep Flash enabled while indiscriminately browsing the web? Very bad things.
What should you do if you use some sites that still require flash? Find a different provider for those services, if possible, who doesn't require Flash for their site. All major browser vendors are abandoning plugins in general, so it's just a matter of time before you'll have to start using some obscure specialty (and/or outdated, and therefore even more vulnerable) browsers if you need Flash.
If you absolutely have no other choice but to use Flash, Chrome currently has a built-in Flash Player that gets automatically updated along with the browser. IE 10 and later, on Windows 8 and later, also has built-in Flash but Chrome is preferred for its (generally) better security and background updates.