You can install MySQL on your workstation, it doesn't need to be on a "server" per se. You still need to use something that can connect to it. From a Java application, for instance, you'd use JDBC; from .Net, you'd probably use ADO.Net; etc.
As far as I know, it will still want to have its server process (mysqld) running and for you to connect to that process via sockets and the like; there's no standard in-process version that I'm aware of. (The server can be listening only on the local interface, though.) There are several alternatives if you want in-process stuff, such as SQLite and HSQLDB.
Of course, if you're feeling really enterprising, there's the open source version of MySQL, which means you could compile it into your app (if you're using C or something that can link to it), but I suspect that's going rather too far. :-)