Level Up Achievements:
When you reach certain levels, you receive an achievement and reward. Note that you can reach that level with any character and any class to get the achievement, since these are tied to your account:

Level 10
Achievement Reward: Banner Sigil (Hellspawn)
Reach Level 10

Level 20
Achievement Reward: Banner Accent (Full Wings)
Reach Level 20

Level 30
Achievement Reward: Banner Sigil (Minion of Hatred)
Reach Level 30

Level 40
Achievement Reward: Banner Pattern (Chevron)
Reach Level 40

Level 50
Achievement Reward: Banner Sigil (Demonspawn)
Reach Level 50

Level 60
Achievement Reward: Banner Shape (Resplendent)
Reach Level 60

Two There Are
Reach Level 60 with 2 characters

Level 60 is the level cap in Diablo 3. For this achievement, you have to reach that level with any 2 characters on the same account.

Warriors of a Broken World
Reach lvl 60 with 5 characters

Heralds of the New Dawn
Reach level 60 with 10 characters

Class Warfare
Reach Level 60 with 2 different classes

You will probably get Class Warfare the same time you get Two There Are, unless you really like to play a single class...

From Arreat to Xiansai
Reach level 60 with 5 different classes

You can get this one along with Warriors of a Broken World if you play 5 characters, each a different class.

Note: All achievements listed on this page are worth 10 points.