In most Console/Handheld Game Companies, it goes like this.
CONCEPTION:
1. Gather a group of people in the company you work in, and assign roles (Programmer, Designer, Artist, etc.) to each person.
2. Get an idea together.
3. Give your Team tasks: Designers write a Game Design Document, Programmers build a Framework, Artists draw Concept Art in 2D, and Producers make a Schedule, Budget Plan, etc.
4. A Designer turns most of this stuff into a Pitch, and go to their Supervisors and/or Publishers (if any). They will give you a Red, Yellow, or Green Light. Red means "NO!", Yellow means "Yes, but change a few stuff", and Green means "Yes".
ALPHA:
5. Now the Programmers try to complete a feature-complete build before Beta, Artists help making the necessary Art (2D and/or 3D), Composers make the necessary sound, and Designers think of how the final product should look and feel like.
6. Show it to the Supervisors and/or Publishers (if any), which will yet again serve you with one of the 3 coloured lights.
It's always a good idea to let your Team test this build before going to Supervisors/Publishers.
BETA:
7. Programmers add any new features to turn it into a complete game, Artists and Composers complete all their Assets, and Designers take the final decisions.
8. As soon as the game got completed, let QA Testers go through the game over 9000 times, make them catch as many Bugs as possible, and let them tell you what they think of your game.
RC:
9. Appoint all the required legal stuff with the Console Makers (Age Ratings, Contracts, Translations, Guidelines, Tests, etc.), and submit the game to the Console Makers (or Publishers, if you're not self-Publishing).
10. Make sure you wake up the Press by announcing it.
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As for the fan games, you should better ask the other people here instead.