The proper way to do a push up?

The way I have been doing push-ups I let my arms come outward but they still become parallel to the ground,but in this video it shows he keeps his arms stay in.

http://video.about.com/usmilitary/Proper-Army-Push-Ups.htm

So am I doing push-ups wrong, or just as long as they become parallel to the ground?

Also they said it was important to keep your head up when doing a push-up as it actually work muscles around the neck too. Is that true?

Thank you.