What exactly happens at the end of a season in Diablo 3?

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.

At the end of the season the character you created becomes a normal character in your list of characters. All items in the season stash will be sent as mail, there is a blue post on this here: http://us.battle.net/d3/en/forum/topic/13922512563

Items will be mailed to you using the same system that delivers Collector's Edition items. You'll have a small envelope button in the lower, left hand corner, and you'll have 30 days to reclaim any items mailed to you in this manner when the Season concludes.

In addition, a new stash tab is being made available in Patch 2.1.0 for players with the Reaper of Souls expansion. If you unlock tabs during a Season that you do not already have unlocked for your non-Seasonal characters, they will rollover to your non-Seasonal characters when the Season ends.

I think if I crammed more Seasons in that sentence, I'd age a year. ;P The end-of-Season process can be a little confusing to explain, but I hope this makes some sense!

And as for the paragon levels and experience earned in the season

When a season ends all of the paragon experience you have earned will be rolled into the appropriate pool of paragon experience for the account and you will be able to start at 0 again in the new season. - Quoted Blue Post by Travis Day

I also found this blog containing answers about seasons ending by a blue user who answers many of the post season questions if this answer doesn't satisfy all of your curiosity. While it is mainly about how it was handled in the ptr, that is essentially the same process that will be used at the end of season 1 now.