Instructions
Minecraft Forge
Navigate to the Minecraft Forge downloads page (link in Resources).
Scroll to the version of Minecraft Forge that's compatible with the version of Thermal Expansion you're downloading and then click "Universal" next to it. (As of August 2013, you need version 7.8.0.688.) Save the file to your desktop.
Launch the Charms search menu, type in "%appdata%" and press "Enter." Open the ".minecraft" folder.
Open your Minecraft Forge downloaded file with a file archiver. In a separate window, open "minecraft.jar" from the .minecraft folder with the file archiver.
Press "Ctrl+A" to select all the files in the Minecraft Forge archiver window and then drag them into the minecraft.jar archiver window.
CoFH Core and Thermal Expansion
Navigate to the Thermal Expansion: Downloads page (link in Resources). Click the "Download CoFH Core" (as of August 2013, version 1.5.2.5) link and then click "Skip Ad." Save the file to your desktop.
Open the Minecraft core mods folder in the .minecraft folder. Drag "CoFHCore.jar" into the folder.
Open the Thermal Expansion: Downloads page again. Click "Download Thermal Expansion" (as of August 2013, version 2.4.6.0) and then click "Skip Ad." Save the file to your desktop.
Open the "mods" folder in your .minecraft folder. Drag the "ThermalExpansion.jar" into the mods folder. Launch Minecraft and start a world to use the Thermal Expansion mod.