The simplest set up is to install separate copies of Filemaker Pro with different install keys on each computer. Leave one computer on all the time with the Files Open by FileMaker Pro, Then the other computers can connect to your database with Open | Remote. Do not under any circumstances connect to the database via opening the file directly via a shared folder.
With this method, you can connect up to 9 computers to the host computer. The data is all stored on the host computer so there is no need to update data on the other computers as they are all accessing the same file.
The next level upgrade is to use FileMaker Server to host your files instead of FileMaker Pro this adds some nice features for managing a hosted database that aren't available to you when you host with FileMaker Pro, but also costs a great deal more money so you should go to the product info pages here and research the features, system requirements and costs to see if you want such an upgrade to your system.