The demo for Monster Hunter: Generation arrives next month but you can play it a few weeks early. If you’re in Europe, then the demo is already publicly available, and you can download it from the eShop with no problems, if you’re in the States, your options are a bit more limited.
Nintendo is sending emails with codes for the demo, though who they send them too is mostly random. Each email has two codes, one for the recipient or one for a friend. Signing up for My Nintendo, the company’s rewards program, is presumably necessary, and making friends with someone else who’s signed up for it can’t hurt.
You can also buy a demo. As part of this week’s E3 Digital Ticket Humble Bundle, a code for Generations is one of the second tier rewards, meaning you’ll get it along with a ton of other goodies for only 4 dollars. If you can’t wait to hunt monsters, than that’s your safest bet.