<p>How exactly do DLL files work? There seems to be an awful lot of them, but I don't know what they are or how they work.</p>

<p>So, what's the deal with them?</p>

What is a DLL?
Dynamic Link Libraries (DLL)s are like EXEs but they are not directly executable.  They are similar to .so files in Linux/Unix.  That is to say, DLLs are MS's implementation of shared libraries.
DLLs are so much like an EXE that the file format itself is the same.  Both EXE and DLLs are based on the Portable Executable (PE) file format.  DLLs can also contain COM components and .NET libraries.
What does a DLL contain?
A DLL contains functions, classes, variables, UIs and resources (such as icons, images, files, ...) that an EXE, or other DLL uses.
Types of libraries:
On virtually all operating systems, there are 2 types of libraries.  Static libraries and dynamic libraries.  In windows the file extensions are as follows: Static libraries (.lib) and dynamic libraries (.dll).  The main difference is that static libraries are linked to the executable at compile time; whereas dynamic linked libraries are not linked until run-time.
More on static and dynamic libraries:
You don't normally see static libraries though on your computer, because a static library is embedded directly inside of a module (EXE or DLL).  A dynamic library is a stand-alone file.
A DLL can be changed at any time and is only loaded at runtime when an EXE explicitly loads the DLL.  A static library cannot be changed once it is compiled within the EXE.
A DLL can be updated individually without updating the EXE itself.
Loading a DLL:
A program loads a DLL at startup, via the Win32 API LoadLibrary, or when it is a dependency of another DLL.  A program uses the GetProcAddress to load a function or LoadResource to load a resource.
Further reading:
Please check
MSDN
or
Wikipedia
for further reading.   Also the sources of this answer.