Very odd, If yo try opening an adobe icon'ed PDF file does it not open Reader? If you don't have a PDF file with the Adobe icon go to 'Get Info' for the file and alter Open with to Adobe Reader. Does it open with Reader?
Are you using the correct user account (admin)? Guest accounts can't install apps. Don't get confused here the users name is not the same as the account type. Goto System Prefs > System 'Accounts'. under My Account see if Admin is under your name. If not change the user to one that does. Then try reinstalling.
You may also have a locked folder where Adobe is trying to install to. make sure the Application folder and any subfolder has admin & system with read & write privileges and everyone else read only.
If all of this is correct you may have a permissions corruption which means you need to run disk repair and you may need to run it from a externally booted HD to get the problem resolved.