As of Minecraft 1.6 texture packs have been officially renamed as resource packs. This guide explains how to install resource packs of all resolutions.
1. Download a texture pack from this website by clicking the Download button on a texture pack page. Do not un-zip the downloaded file.
2. Open Minecraft and click Resource Packs from the settings menu.
3. Click Open resource pack folder. This opens up the folder where the texture packs are stored. If the folder does not open you will need to manually locate it:
4. Place the texture pack in the folder leaving the file as a .zip.
5. The texture pack will now show up in Minecraft. Simply click the pack and it will be applied to Minecraft!