Generic mana cost
A generic mana cost is a mana cost that can be paid with mana of any type; meaning mana of any color, as well as colorless mana.
A generic mana cost is represented by a number or an X in a gray circle (
, 
, 
, 
, 
, and so on).
Difference with colorless
There is no such thing as "generic mana". [1] Though you can use colorless mana to pay for generic mana costs, they are not the same thing. "Generic" is a type of cost. "Colorless" is a type of mana. You can't produce generic mana, but you can produce colorless mana and have a colorless cost (first introduced with Oath of the Gatewatch).
R&D came to the conclusion that they had to stop using the mana circle with a number to represent both generic and colorless mana. Changing the representation of colorless mana was easier, as it's represented far less, so they to made the new mana symbol 
.
References
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