Editor service awards are simply a way of acknowledging the level of contribution of editors based on specific benchmarks defined by the number of contributions and the length of service and to provide for editors who wish to display a mark of their level of service based on these benchmarks.
Unlike other awards, editor service badges are achieved strictly by a mechanical count of time of service and number of edits. There is no process for receiving the awards - you just pick the badge up yourself if you're eligible. This is one award that is intended to be given to yourself. (Of course it can also be given to an eligible editor by a second party.) If you prefer, you can think of it as automatically assigned, like credit for a "DYK?", or the years-of-service bars that appear on (usually) the lower part of the left sleeves of many police, military, etc., uniforms.
Please remember that time spent with and number of edits to Wikipedia are not indicative of the quality of an editor's contributions or of their diplomatic ability. "Veteran" editors have no more authority than "novice" editors.
"Tutnum", etc., are alternative terms for those who find "yeoman editor", "senior editor" etc. too stuffy. For the sake of variety, three "badge" variants (a medal, a book, and even a small ribbon, thanks to Wikipedia:Ribbons) are provided for each level. Editors may display the one of their liking, or for that matter a combination (or none). Continuing the "Tutnum" vein, "Burba" is Italian for "Rookie", according to Babelfish; "Novato" is Portuguese for "Beginner"; "Grognard" ("Grumbler" in French) was Napoleon's pet name for his Old Guard; and "Tutnum" is just a made-up word that sounds impressive.
You can get your edit count here (In case you have forgotten, this will also show you the date of your first edit (logged in with a real user account), or alternatively you can click my preferences and your edit count will appear there. (Although that number is slightly larger because it includes deleted contributions).