Is Flash Player only for Safari? Do I need Flash Player for Firefox?

The download from Adobe's website is for multiple browsers.

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A Flash tip to help avoid security issues later: There are known scammer who fake a "Flash is out of date" pop-up that leads you to a site that installs malware or can compromise your private data. If you see a pop-up in FF saying you need to upgrade Flash, the only safe thing to do is go to this official Adobe site:

https://get.adobe.com/flashplayer/

and follow the downloads instructions there. It will show the currently available version. You can quickly tell what version is on your computer by doing System Preference > Flash Player and selecting the "Updates" tab.

Do NOT choose to update Flash from any pop-up. Here are several articles with more info on the "Fake Flash" issue:

http://www.thesafemac.com/?s=flash&submit=Search

Fake Flash pop-ups commonly contain grammatical or spelling errors. One common one is "Failure to install this update may cause your computer to crush." Using only the official Adobe site I linked plus checking your current version via System Preferences will avoid your having to figure out if a pop-up is fake.