LAMP is the acronym commonly applied to a set of popular open source platforms and tools that combine to support Web services:
Slashdot, Freshmeat, Apacheweek and Perl.com are all LAMP-based. In the weblog world, The Blogging ecosystem, Organica, Daypop, and Technorati are all Web services built on a LAMP foundation. The combinarion of an open, stable, scalable, and robust combination of Operating System, web server, database, and scripting language make it really easy to create and update dynamic web content -- using systems that are relatively easy and cost-free to create and improve.
Linux at the core gives a stable, responsive operating system that can be well secured for the internet, and that also provides extremely robust networking features, important as you scale a backend from one machine to many. Apache, the market-dominating web server, is also extremely fast and stable. Apache also allows for integration of the scripting language through its modular extensions system -- mod_perl for Perl, mod_python for python, and libphp4.so for PHP4. This means that the scripting language interpreter is built into each apache executable, and page processing time is very fast. Scripts can also be precompiled and run straight from memory, alleviating somewhat the performance hit that scripting languages usually bring, while allowing the programmer to code using a higher-level language rather than a compiled language like C or C++.
MySQL is one of a number of fast SQL databases available for Linux users, and is widely used for Web applications due to its speed and integration with scripting languages. Bindings exist for MySQL in PHP, Perl, and Python, as well as for Java and a host of other languages. While other open source databases are gaining in popularity (notably PostgreSQL), none has yet to match the popularity with web developers as MySQL. An interesting sidenote here is that many people do switch out databases at later stages in a project. For example, Sourceforge.net started out as a LAMP project, but switched to Oracle (and now DB2) as the backend database with only minimal rewriting of the core code.
The "P" in LAMP stands for a number of scripting languages, including Perl, Python, and PHP, which are the most popular three. But Java, Ruby, and Tcl are also popular alternatives. A significant advantage with all of these languages is a well-established base of existing open source code to access and manipulate databases, parse and create HTML and XML, perform cookie and session management, and so on.
Add all of these tools together, and you have an open source application serving powerhouse that outclasses any combination of proprietary alternatives.