What are .sln and .vcproj files, and what do they contain?

A project file .vcproj / .vcxproj contains settings on how to compile your code into a DLL or a binary file, or something else that the linker can assemble into one unit. A project file is just an xml file that contains compiler settings, linker settings, and describes which files you want to compile.

A solution file *.slnis a text file that groups together multiple project files.

So if you think of it like a tree, then you have got a good mental picture of it like this:

.sln
   .vcproj
      .h
      .h
      .cpp
      .cpp
   .vcxproj
      .h
      .h
      .cpp
      .cpp
   .csproj
      .cs

Solution files and project files are in an XML format and describe the parts of your projects and their relations, configurations and so on. In fact, both of these files are simply MSBuild scripts (which are run through MSBuild when, you guessed it, building your project.)

This means they are easy to manipulate by hand if needs be (though this should be a rare case) and also allows to add custom parts to the build script, create custom build scripts for MSBuild that can include the solution file, among other things, or just simple auto-build scripts that pass the solution file (or project) to MSBuild, say, on version control check-in.

The difference between solution files and project files is that a project file holds information specific to that project, unaware of its solution (though, Visual Studio will look up the hierarchy to an extent in an attempt find the relevant solution when opening a project, if one exists); the solution file is aware of all projects that are part of that solution and references each of them (such as a directory of files, if you like, but with projects), it also contains solution-wide information / configuration, that can be applicable to all projects within the solution.

As pointed out by Hans Passant, there is an exception: files for C++ projects pre-VS2010 are not XML MSBuild files, but are instead a format documented by Microsoft on MSDN.

In short: one is for solution, and the other is for project, and a solution can contain multiple projects.

A .vcproj file contains information about HOW to compile source to a target (mostly, an executable). In many cases, it is crucial to have the project file for successful compilation, so do not delete it. It is compareable to a .dsp file (Visual Studio 6), a .prj file (Borland compilers), or a Makefile (Unix, GNU compilers) and contains paths and compiler/linker command-line options.

A .sln file is merely a collection of multiple .vcproj files. As Visual Studio can automatically create one if not present, there is no need to keep it for distribution or archiving. It's the successor of a .dsw file (Visual Studio 6). Its name "Solution file" is IMHO misleading.