Replace Apache completely
If you don't need all the features of Apache, simply replace it with something more scalable. Currently, the best options appear to be servers that use a non-blocking I/O technology and connect to all clients with the same process. That's right - only ONE process. The best include:

thttpd - http://www.acme.com/software/thttpd/
Caudium - http://caudium.net/index.html
Roxen - http://www.roxen.com/products/webserver/
Zeus ($$) - http://www.zeus.co.uk

Solutions: PHP's CPU & RAM Usage
Compiling PHP scripts is usually more expensive than running them. So why not use a simple tool that keeps them precompiled? I highly recommend Turck MMCache. Alternatives include PHP Accelerator, APC, & Zend Accelerator. You will see a speed increase of 2x-10x, simple as that. I have no stats on the RAM improvement at this time.

Solutions: Optimize Database Queries
This is covered in detail everywhere, so just keep in mind a few important notes: One bad query statement running often can bring your site to its knees. Two or three bad query statements don't perform much different than one. In other words, if you optimize one query you may not see any server-wide speed improvement. If you find & optimize ALL your bad queries you may suddenly see a 5x server speed improvement. The log-slow-queries feature of MySQL can be very helpful.

How to log slow queries:

# vi /etc/rc.d/init.d/mysqld

Find this line:
SAFE_MYSQLD_OPTIONS="--defaults-file=/etc/my.cnf"

change it to:
SAFE_MYSQLD_OPTIONS="--defaults-file=/etc/my.cnf --log-slow-queries=/var/log/slow-queries.log"

As you can see, we added the option of logging all slow queries to /var/log/slow-queries.log
Close and save mysqld. Shift + Z + Z

touch /var/log/slow-queries.log
chmod 644 /var/log/slow-queries.log

restart mysql
service myslqd restart
mysqld will log all slow queries to this file.

References
These sites contain additional, more well known methods for optimization.

Tuning Apache and PHP for Speed on Unix - http://php.weblogs.com/tuning_apache_unix
Getting maximum performance from MySQL - http://www.f3n.de/doku/mysql/manual_10.html
System Tuning Info for Linux Servers - http://people.redhat.com/alikins/system_tuning.html
mod_perl Performance Tuning (applies outside perl) - http://perl.apache.org/docs/1.0/guide/performance.html

Once again, if this has any errors or important omissions, please let me know and I will correct them.
If you experience a capacity increase on your server after trying the optimizations, let me know!

Article written by spagmoid additions by: albo,huck on the ev1 forums.