Amber is commonly mined from "Amber-Bearing Stone", but is also occasionally dropped by Furious Zombies. This ore is found in single blocks scattered throught the underground. It is more common at higher elevations, and can often be found exposed at the surface, even well above sea level. The ore requires an iron pickaxe or better to mine; normally it drops a single chunk per ore block, but the Fortune enchantment can increase this. If obtained with Silk Touch, the ore blocks can also be smelted.
Amber chunks can be crafted 4:1 into Amber blocks, which in turn can be crafted 4:4 into amber bricks. Both amber blocks and amber bricks can easily be crafted back into 4 chunks of amber. The chunks and occasionally blocks are used in a number of crafting recipes, and are also a prominent source for Vinculum essentia.
In 18.104.22.168, there is a troublesome bug, where surrounding an amber block or bricks on all sides with opaque blocks (e.g., burying it in the ground) can crash the world, and thereafter cause it to crash on startup. The only remedy for this bug is to edit the save files with MCEdit (or an NBT editor capable of handling blocks) to delete the offending block, and/or the blocks surrounding it.