I would also recommend
PostgreSQL as the server. It scales very well, offers very good performance, has offered a TON of features for a long time, and is 100 percent free of charge (it's truly open source.)
If the anticipated database size won't exceed 2 GB, MSDE should work well for you also. It is also free, and is 100 percent compatible with SQL Server if/when you need to scale up.
Note that neither of these database offer any sort of performance guarantee. They're free, after all. However, if you are currently using Access/Filemaker, You wouldn't need to worry about this. Even MSDE (which
is SQL Server, throttled to approximate Access' multi-user performance) is a step up, bercause it won't get corrupted the way Access can with multi-user apps. And, it offers true database server features like stored procedures, triggers, and real security.
You would install the database server on a dedicated server. (the dedicated part isn't required, but it makes sense to do so.) The data is stored to locations defined by the server (it can be on server local drives, network drives, or a combination. Typically, on SCSI/SATA drives with RAID.
SQL commands are sent to the server, and the database service intrepets the SQL to store/edit/retrieve the data in the locations defined by the database configuration. How you actually access the database can cover a wide range. As you mention, Access as a front end, with linked tables and passthrough queries. (if you don't use passthrough queries, there's a lot of network traffic involved, as query processing is handled on the Access side.)
Or, you could use a programming language (
VB/
VB.Net/C#/C++) to write your apps specifically matched to your requirements. More work, but, you have full control of everything on the front end.
Or, you could use a web server, and build an ASP/PHP middle tier and use web browsers to access the data.