First, you need to find the YouTube video you want and click the download icon on the thumbnail or on the video player.
A window will open up asking you to select download format. If the video is available in High Definition MP4 and your device is capable of playing it - choose this one. Otherwise choose the High Quality MP4 version
A "Save as" box will pop up asking for location to save the file. Make sure you click "Save" not "Open"
If you are using Internet Explorer you might see the Yellow Information Bar showing. If you do - right click it and select "Download File" or right click the "Download" link on the box and choose "Save target as..."
Once the video is downloaded on your computer connect your iPhone/iPod and start iTunes. From the "File" menu click "Add File to library..."
Browse for the downloaded file. It is usually named "video.mp4", so you might want to rename it. Select the file and click "Open"
The video should appear in your Movies library now. From here you can play it and adjust details like name, artist and etc
Select your device on the left, click on the "Movies" tab at the top and check "Sync Movies". Make sure the video you downloaded is selected and click "Apply" at the bottom. Your video will now be transferred on the device