Go to your gamedir, and edit ra2.ini. Find these lines:
[Video]
ScreenWidth=640
ScreenHeight=480
StretchMovies=no
Edit the values of ScreenWidth and ScreenHeight to set your resolution. Note that RA2 uses 16 bit color, and if your system can handle a specific resolution at 32 bit color but not 16 bit color, RA2 will not run at that resolution.
As far as I can tell, setting StretchMovies to yes does absolutely nothing.
If you have Yuri's Revenge, do the same for ra2md.ini.
The game view is pixel-based. The more pixels in your resolution, the more territory you can see at once. For example, 1280x800 will appear to be wider and a little bit taller than 1024x768, but 1280x1024 will appear equally wide as 1280x800 and taller. The main menus are centred while the in-game menus display along the edges of the resolution. Everything scales correctly.
Widescreen in multiplayer works as described above. However, many hosts enforce resolution limits. If you try to join a game with a resolution limit, you will be forced to play the game at a non-widescreen resolution (usually 640x480). If you host a game and are using a widescreen resolution, you will not have the option of enforcing a resolution limit.