Lots of comments about FileMaker being non-standard. But what is "standard"? By "standard", many people mean that a database supports Structured Query Language (SQL) (ISO Standard 9075) and FileMaker has and continues to support SQL. How every database engine supports SQL is proprietary to every database. Now it might be open source such as MySQL, but SQL is a standard to support, not the underlying language of how it is accomplished.
When most people talk about databases, they are only talking about the backend tables and schema. The front end user interface is frequently something else. And most of them now render those results as html pages via open standards like PHP. Again, FileMaker fully supports PHP calls and Apache or IIS (depending on which OS platform you are on).
So I would disagree with people saying FileMaker is non-standard.
What is unique about FileMaker is its tight integration between the schema and the User Interface. This is similar to Apple's tight integration between hardware and the Operating system, which has some nice benefits. Interestingly, FileMaker is owned by Apple, but I guess that is another topic.
Generally, FileMaker's User Interface is considerably easier to use than most open standards and most people stick to FileMaker's client User Interface instead of web interfaces. There are still a number of things supported only in FileMaker User Interface that can't be duplicated in a web browser.
FileMaker really makes rapid application development much easier with its close integration of schema and user interface. This makes development cost a whole lot less in most cases.
FileMaker's database services can be spread among up to 3 machines giving it primitive load balancing abilities with web services. While FileMaker easily supports hundreds of users, if you go into thousands of simultaneous users, many SQL only databases (eg Oracle, MS SQL Server, MySQL, Postgres) are designed to better spread out the load across more machines. Basically, if you have high simultaneous transactions, FileMaker is not your solution. For example, a company with many point of sale terminals from all over the county hitting it at the same time.
While FileMaker supports SQL and PHP, using it only that way is a waste of the money spent on the license for the FileMaker User Interface. It would not be a cost effective solution to develop a web front end and pay the full FileMaker license cost for only a backend. So, FileMaker's support of PHP and SQL is best combined with companies that have an in-house solution for staff, but also want to integrate that with their web development team for outside customers.
One last note is that FileMaker's tight integration of schema and User Interface makes security much easier. Obviously you have to set up the groups and users and I usually integrate FileMaker with Active Directory (or Open Directory). But when you use the FileMaker Client and Server connections, turning on encryption security is a single checkbox on the server. FileMaker handles all of the certificates and uses an AES 256bit cipher (at least since version 11, maybe before then too). Currently, the US Government considers that approved for up to and including the first level of Top Secret communications. In typical SQL systems, there is a lot of work to configure security on the database end as well as the user interface end of things and it is much more work than a single checkbox.
FileMaker's target audience has been small to medium sized companies, usually with 5 to 200 users, and it is a well priced product for rapid application development of databases for companies of that size.
And I can't end this comment without commenting on how easy it is to create and deploy a mobile solution on iOS devices like iPads and iPhones. FileMaker Go is a free app for use on these mobile devices and they fully support the same user interface and security. In fact, I am aware of one company that uses FileMaker as a front end interface for their Oracle database simply for access on iPhones. Expect a lot more in the mobile market in the future and FileMaker is clearly targeting mobile users.