Q1: What is Seasons?

Seasons is a new game mode introduced to Diablo III that allows you to relive the leveling experience of your character to reap new rewards such as season-only legendaries, special transmogs and other bonuses. You will have to start from scratch, that means:

• Seasonal stash and storage is separate from Normal (you have nothing in your Seasonal stash)
  
• Artisans are also separate from Normal (they start from level 1)
  

You will however not be given additional character slots to participate in Seasons.
Q2: What are the requirements to join a Season?

You need to create a new character whenever a Season starts. When creating the new character, check the box where it says Seasonal Hero and your character will be available for play during that season.
Q3: Is there a Hardcore mode for Seasons?

Yes there is.
Q4: What's so great about Seasons? Why should I play it?

Seasons gives players a couple of benefits, namely:

• Your chance at scoring Seasonal Legendaries before they roll over to Normal's loot pool
  
• Exclusive transmog (the look of your gear) upon reaching level 70 with a Seasonal Hero (per Season)
  
• Unlocking Season-exclusive Achievements
  
• Earn yourself in a place on the Seasonal Leaderboard (leaderboard for class, etc.)
  

Also, all progress you gain from Seasons will roll over to Normal servers. What this means is that all XP gained for your character(s) on Seasons will be added into your Normal account. Gears and loot found from your Seasonal Hero(es) will remain on them and those stored in your Seasonal stash will also be mailed to you.
Q5: I want to play my normal characters, can I do that?

Yes. At any time, you can play your Seasonal Hero as well as your Normal Hero. Any progress made between these 2 characters are separate and saves accordingly in their respective servers.
Q6: When does a Season start?

You can find the most updated Season duration from here.

Current Season: 2
Duration of Season: 13th March 2015 - 6th April 2015, 12.00 p.m. (times are in GMT+8)
More info on Season 2 at here.
Q7: Do I need to create a new account to participate in Seasons?

No. You only need 1 account for this.
Q8: May I have another player with Normal hero play with my Season hero?

No. Seasonal Heroes may only interact with other Seasonal Heroes.
Q9: What happens to my Seasonal Hero(es) when a new Season starts?

Your previous Seasonal Hero(es) will become a non-Seasonal hero(es). To participate in the on-going Season, you have to create a new Seasonal Hero.
=========================

According to the official post, Season 2 for Seasons will end on the 6th of April 2015 for US servers.

Sneak peek at what to expect for Season 3 can be found here.

Booyah!

More details:

Season 1
Announcement thread for Season 1
Roundup for Seasons on Diablo III
Season 1 Exclusive Legendaries
Announcement on the End of Season 1

Season 2
What to Expect for Season 2
Announcement for Season 2
Announcement on the End of Season 2