We're only just starting the very first Season, but it's important to know what happens to your progress at the end. When a Season concludes, your current Seasonal heroes will become non-Seasonal. This means that if you were playing a Seasonal Hardcore hero, they'll instead become a regular Hardcore hero.
Your heroes keep all their gear, crafting materials, gold, blood shards, Paragon experience, and any new recipes they may have earned during the course of a Season. This also includes any new Stash tabs purchased or Artisan levels earned (if higher than your non-Seasonal profile). Finally, if you've managed to earn achievements during a Season that you've never conquered before, you'll earn them in the corresponding non-Seasonal mode.
While you'll be starting from scratch when a new Season begins, your non-Seasonal heroes will reap the rewards with new items, experience and achievements!