Most SQL database engines are implemented as a separate server process. Programs that want to access the database communicate with the server using some kind of interprocess communication (typically TCP/IP) to send requests to the server and to receive back results. SQLite does not work this way. With SQLite, the process that wants to access the database reads and writes directly from the database files on disk. There is no intermediary server process.
There are advantages and disadvantages to being serverless. The main advantage is that there is no separate server process to install, setup, configure, initialize, manage, and troubleshoot. This is one reason why SQLite is a "zero-configuration" database engine. Programs that use SQLite require no administrative support for setting up the database engine before they are run. Any program that is able to access the disk is able to use an SQLite database.
On the other hand, a database engine that uses a server can provide better protection from bugs in the client application - stray pointers in a client cannot corrupt memory on the server. And because a server is a single persistent process, it is able to control database access with more precision, allowing for finer-grained locking and better concurrency.
Most SQL database engines are client/server based. Of those that are serverless, SQLite is the only one known to this author that allows multiple applications to access the same database at the same time.
(This section was added on 2018-04-02)
Recently, folks have begun to use the word "serverless" to mean something subtly different from its intended meaning in this document. Here are two possible definitions of "serverless":
Classic Serverless: The database engine runs within the same process, thread, and address space as the application. There is no message passing or network activity.
Neo-Serverless: The database engine runs in a separate namespace from the application, probably on a separate machine, but the database is provided as a turn-key service by the hosting provider, requires no management or administration by the application owners, and is so easy to use that the developers can think of the database as being serverless even if it really does use a server under the covers.
SQLite is an example of a classic serverless database engine. With SQLite, there are no other processes, threads, machines, or other mechanisms (apart from host computer OS and filesystem) to help provide database services or implementation. There really is no server.
Microsoft Azure Cosmo DB and Amazon S3 are examples of a neo-serverless databases. These database are implemented by server processes running separately in the cloud. But the servers are maintained and administered by the ISP, not by the application developer. Application developers just use the service. Developers do not have to provision, configure, or manage database server instances, as all of that work is handled automatically by the service provider. Database servers do in fact exist, they are just hidden from the developers.
It is important to understand these two different definitions for "serverless". When a database claims to be "serverless", be sure to discern whether they mean "classic serverless" or "neo-serverless".