There are currently 3 Flash plug-in variants in existence, each of which works with different browsers. As @leocg notes, the ppapi version works with Chrome-based browsers like the current Opera; the active-x (or 'aex') version works with IE; and the npapi version works with Firefox and Olde Opera. And yes, you can (and may need to) have all versions successfully installed on the same computer - I do. When Flash updates its version number, you may need to update each variant separately (unless one or any are set to auto-update).
