How do you improve Drupal performance?
Caching, caching, and caching.

Some suggestions I have previously given to a similar question on d.o.


Putting Varnish or another reverse-proxy in front of your http-deamon is probably the single best thing you can do.
During DrupalCon Copehagen, Rasmus stated that using a php opcode cache, such as APC, is one of the best things you can do to speed up PHP in general. Performance improves with newer versions of PHP. There is also additional benefits to upgrading PHP when you upgrade Drupal. From 6 to 8, Drupal will go through a major shift towards object orientation, which is also were most performance improvements happens in the newer PHP versions.
Memcache is a popular choice for speeding up cache, by putting the cache into memory instead of disc.
Panels + Caching combined with Cache actions can increase performance significantly, even for logged in users, as it supports quite complex logic.
The Entity Cache is a nice and zero-conf speed boost for anyone using Drupal 7.
Write-heavy sites have fewer "well established" solutions. Some options include.

Moving frequent writes entirely, for example statistics, somewhere else, such as google analytics.
Caching frequent write operations with a custom solution in something like NodeJS that will write to DB once every Xth second.
Sacrifice the sacred ACID, and use a database like MongoDB. (See Berdir's comment below)
Cluster your SQL-database. Do reads from one database, writes to another. This is native to D7 and Pressflow can help with that in D6.