It is a collection of the many objects we program with in .NET.  Some of those objects are, for example, the textbox objects that know how to parse strings, the button objects that know when they are pressed and how to center an icon image, and the "common dialog controls" which help you browse the directory or choose colors.  Other objects do Just-In-Time Compiling of the .NET intermediate language and perform all sorts of other vital operations.

An official 'techy' Microsoft page on what the Framework is.

 Download the .NET Framework v 2.0 (about 24 MB) and then run the dotnetfx.exe to install it

• Why is the framework so big?  Can I install just the pieces I need?

• The framework is so big, because currently Microsoft is distributing it in an all-or-nothing fashion.  There are many, many new and potentially useful objects that Microsoft wants developers (themselves included) to have use of in Windows. 
  The good news is this means Microsoft is including the framework with most all new software they are distributing including new operating systems.  The bad news is that means it pretty much adds almost 20 MB to to size of your setup for the time being.  (On the other hand, most programs these days are distributed on a CD which generally holds between 650 and 800 MB of information.)

•  What operating systems already have it installed?  Which operating systems can I get my programs to run on if .NET framework is installed?

• While none of the currently shipping OS's include the framework, future ones will.  That said, anybody who has their Windows Updates up-to-date probably has it installed and many new computers come with the update patches already installed.  Basically any 32-bit Windows version can use .NET (Win 9x, ME, NT, 2000, XP [everything except Win 3.11])

• Can't I just include this in the setup that I create for my program?

• No -- that would be the easiest solution but recently Microsoft modified their installer to discourage programmers from doing that. They'd like to be in charge of when and where somebody gets the .Net Framework.  You CAN still download the .Net Framework installer (about 24 MB -- see above) and include it as a separate installation on your CD.  Or you can encourage your users to go to Windows Update and get it (that's what MS suggests).