Exporting a Master File creates a completely self contained movie that can then be either put into another project, turned in, or posted to Youtube.
If you are turning the video in or posting to Youtube, you should export with h.264.
[Note: If you are only sending your file to Youtube you should Share it directly to Youtube with File > Share > Youtube ]
1. Name your Project! This name appears at the top of the video! "Untitled Project" means you don't know what you are doing and will piss me off.
2. Select the project. The project is the thing in the picture above. This makes it so you export your project and not some random clip that was the last thing you clicked.
3. File > Share > Master File
4. Change the Settings so the Video codec is H.264
This makes a nice small video file that can be put on a thumb drive, uploaded to Youtube, or turned in to the server.
ProRes 422
If you are adding your video to another project like After Effects or another Final Cut project, you should export with the Apple ProRes 422 codec. ProRes is a very uncompressed codec that will be a very large file but will not lose clarity or detail if it is recompressed later.
You want to avoid compressing with H.264 and then using that file in another project and then compressing again with H.264. Your computer may lag and the quality goes down with each compression. That's what ProRes is for.