In a nutshell, and based on this wiki page:

Heroes

At the end of a Season, your Seasonal Normal (or Hardcore) heroes, inventory, shared stash, and Paragon experience will transfer to your Normal (or Hardcore) non-Seasonal profile. Any items, equipped or in the hero's inventory, will roll-over with the hero itself.
Heroes seasonal tags are removed.

Shared Stash

Items in your Seasonal heroes' shared stash will be mailed to your Normal/Hardcore non-Seasonal profile. These items can be retrieved by entering a game on any non-Seasonal hero and clicking on the mail icon located in the lower left of your screen.
These items aren't automatically placed in your non-Seasonal stash, so you can wait and make room first before claiming them (if needed). Under current design, mailed items from your Seasonal stash will remain available for 30 days after a Season has ended.

Paragon Experience

Paragon experience earned during a Season will transfer over directly without any additional involvement required from players.

Achievements

If you unlock an achievement on a Seasonal hero that you haven't already unlocked on a non-Seasonal hero, you'll gain credit for both at the same time.

Gold and blood shards

Gold will be automatically transferred from your Seasonal profile to your non-Seasonal profile. Ditto for Blood Shards.
When transferring over your Blood Shards from your Seasonal profile to your non-Seasonal profile, the 500 cap will temporarily be ignored. In this case, players will be unable to pick up any more Blood Shards until they drop below 500.

Artisan levels and learned crafting plans

If your Seasonal Artisan is higher level, its level should transfer at the end of the season. Any recipes you have on your Seasonal Artisan that are not on your non-Season Artisan are transferred as well.

Legendary gems

Under current design, if you drop a Seasonal legendary gem that you already possess in Normal mode, you'll end up with two of such gems.

Rewards

As stated by the in-game help screen, you'll obtain exclusive Transmogrifications if you reach the level 70. It's also possible to unlock exclusive Banner styles.

Since this has changed over the past several seasons,

Any currency, materials, recipes, items, Paragon experience, and Artisan progression earned during a Season will be rolled over to a player's non-Seasonal profile once the Season concludes. (src)

The Kanai Cube perks are included as part of Artisan progression. (src) And any other progress that exceeds that of your regular account is imported as well (gold, shards, artisan recipes). The character and their inventory is simply converted to a regular character, and the seasonal shared stash is "mailed" to your regular stash, whereupon you'll have a month to reconcile those items into your regular stash's limits.

Paragon levels are converted to raw experience points and added to the regular account's Paragon. (src) You can use this calculator if you want to predict your final level.