Minecraft Forge is a basically a mod development kit and framework for Minecraft.
If you are a user, and not a mod developer, all you need to know is how to install the right Forge for your Minecraft version, and how to install mods once you have Forge. Okay, and I'll go over how to download the Forge version as well.
Recent versions of Forge have a very, very nice java installer. First, go to the download page at the Forge site, http://files.minecraftforge.net. Wow, they have just changed this page around significantly. Looks like you scroll down to where the Minecraft versions are listed, then select your version, then the page reloads, then you can click the Installer-Win icon for Recommended or Latest. I recommend Recommended.
On the next page, don't be fooled by the stuff on the page, just click Skip in the upper right corner.
This should start the download however that works for your browser. You should save the file, then open it when the download is complete.
When you run the installer it will look something like this:
After a short time it should complete:
At this point, load the launcher and create a new profile.
And now your launcher should allow you to select the new version with Forge, like so:
Before you try to add a mod, though, you'll want to startup the profile first. Make sure the profile is selected in the launcher and click Play. Then wait for Minecraft to load. When it does, you should see the confirmation that Forge is active in the lower left corner:
The simple way to find the directory is to go back to the launcher, Edit Profile and click the Open Game Dir button. You will see a list of directories, including the new "mods" directory. Just copy your mod's jar file there and hit Play and that's all it should take.