COM is an acronym for Component Object Model; it is an object orientated layer (and associated services) on top of DCE RPC (an open standard) and defines a common calling convention that enables code written in any language to call and interoperate with code written in any other language (provided those languages are COM aware). Not only can the code be written in any language, but it need not even be part of the same executable; the code can be loaded from a DLL, be found in another process running on the same machine, or, with DCOM (Distributed COM), be found in another process on a remote machine, all without your code even needing to know where a component resides.
There is a subset of COM known as OLE Automation which comprises a set of COM interfaces that allow loose binding to COM objects, so that they can be introspected and called at run-time without compile-time knowledge of how the object works. The PHP COM extension utilizes the OLE Automation interfaces to allow you to create and call compatible objects from your scripts. Technically speaking, this should really be called the "OLE Automation Extension for PHP", since not all COM objects are OLE compatible.
Now, why would or should you use COM? COM is one of the main ways to glue applications and components together on the Windows platform; using COM you can launch Microsoft Word, fill in a document template and save the result as a Word document and send it to a visitor of your web site. You can also use COM to perform administrative tasks for your network and to configure your IIS; these are just the most common uses; you can do much more with COM.
Additionally, we support the instantiation and creation of .Net assemblies using the COM interoperability layer provided by Microsoft.