Is Adobe Flash Player safe to download?

Flash Player is is cesspool of security issues and a major cause of instability on your Mac. Avoid if at all possible.

Not that I would ever disagree with Terence, BUT:

The latest version of Adobe FlashPlayer can be obtained from here:

http://www.adobe.com/shockwave/download/download.cgi?P1_Prod_Version=ShockwaveFl ash

You can check here: http://www.adobe.com/products/flash/about/ to see which version you should install for your Mac and OS. Note that the latest Flash Player update only works on YouTube with Safari in 32 bit mode - until YouTube correct this.

You should first uninstall any previous version of Flash Player, using the uninstaller from here (make sure you use the correct one!):

http://kb2.adobe.com/cps/865/cpsid_86551.html#prob1=uninst,os=m10.6,

and also that you follow the instructions closely, such as closing ALL applications first before installing. You must also carry out a permission repair after installing anything from Adobe.

If Flash Player is so bad, how do I safely play the content it's called in for?

You'll need to use Flash. Terence posted to avoid if possible. Make sure you only download it from the links Klaus 1 provided.

Personally, I avoid that content.