You have multiple options depending on your programming skills (or that of a consultant you hire) and the software you have acquired. I'm not going to spell out all the technical details here, just list the basic options available. You'll need to research them in more depth to decide on which route you want to take.
You can use custom web publishing with PHP to serve the database up to the web with password controls to limit access. This option is the most work, but is the most flexible if your remote users do not have their own copies of FileMaker and you want them to use the web browsers to access the database.
You can use instant web publishing to make your database available to people using a web browser to access your database. This takes less work and does not require a knowledge of PHP, but also limits many interface options so you have to adjust your existing database layouts (or create layouts just for web use) to work within those limitations.
If your users have copies of FileMaker Pro, they can use remote access to connect directly to your database over the internet. This is often called a WAN or Wide Area Network. Connection speeds are generally quite a bit slower than over a Local network, but no web publishing is required in order for you to set this up.
With all three options, you can either host the database yourself. (Computer, firewalls, ports, and internet router need correct settings to make this happen.) Or you can hire a hosting company to host the database for you.
Thanks Phil, I gives me food for thought.