FileMaker 12 cannot open files from earlier versions without converting them to .fmp12 format. Earlier versions such as FileMaker 9 cannot open FileMaker 12 files--the file format has changed.
You have the choice between buying two copies of FileMaker 12 and converting your files, or making a "downgrade" purchase (I think this is still possible) of FileMaker 11. FileMaker 11 and FileMaker 9 can access the same files.
However you decide to do that, you can then open the file with FileMaker Pro on the computer where the file is located and then the second computer launches FileMaker and uses Open Remote to find and open the database.
Do NOT under ANY circumstances, put the file in a shared directory and try sharing it from that directory by opening the file directly. THis can damage your file.
Cloud type services can be a useful way to keep a backup copy of your file offisite, but is not a practical means of sharing the database for access by multiple users.