You first have to get a referral from an engineering trainer. There are a number of people who will give you a referral. Remember, you need 200 engineering skill and must be at least level 20 to be eligible.
The person you are sent to (either Tinkmaster Overspark, or Oglethorpe Obnoticus, depending on who referred you) will hand you an unsigned pledge, and ask that you sign and return it. This seals your commitment to Gnomish Engineering.
Having done that, you are given a new ability called Gnomish Engineer in your spellbook (Clicking opens Engineering dialog), which entitles you to training on the 'gnome engineer only' engineering recipes. Congratulations! You have specialized in Gnome Engineering!
Membership
Having completed the exercises above, you used to be given a Gnome Engineer Membership Card. This remains valid for 14 days of 'played time' (336 hours in-game), not 14 real days.
Prior to Patch 3.1.0, you could renew this card for 2 gold, and as a side benefit get a renewal gift in the mail, which had a chance of being a useful engineering schematic. The schematics [Schematic: Lil' Smoky] and [Schematic: Pet Bombling] were only acquirable this way.
Now, however, it is simply an ability in your spellbook, called Gnomish Engineer. Clicking it opens the Engineering dialog.
Gnome Engineering items
Gnome Engineering items require specialization in Gnome Engineering to craft.
Since patch 3.2.0, Gnomish Engineers and Goblin Engineers can now switch profession specializations for a fee.
You can now switch specializations by visiting Narain Soothfancy in north east Tanaris. The book on the table in his hut will allow you to switch specializations for a fee of 150g.
Note that switching specializations will cause you to unlearn all associated recipes for your current specialization.
Notes
The measurements Gnomish engineer's and Gnomes in general use is revealed by Fizzle Brassbolts in the quest Martek the Exiled. These measurements are: