MySQL performance is a large and complicated topic. You can find many books on the topic and no answer here can hope to touch on all the important issues.
From the perspective of data base design, the main steps (not necessarily in order) are to design a good schema, define the right kind of indexes to support the queries you expect to process, pick the right engine for each table, and design the queries themselves for best performance. The MySQL Query Analyzer is a tool for addressing these issues, as is the EXPLAIN statement.
In terms of server configuration, there's a huge amount of tuning that goes into improving performance. As the MySQL manual topic Estimating Query Performance describes, the fundamental unit of performance is a disk seek, and theoretically this is log N in the number of rows. However, both the underlying OS and the MySQL server may do quite a bit of caching, which can make the actual performance grow much more slowly in practice (not to mention making it faster in absolute terms). Follow the links from the above page, search for some server configuration guides on the web, and/or read a book or two to get some insight into how to use all the performance-related parameters that MySQL provides.