Proof of Work cryptocurrency mining requires a processor to perform calculations called hashes. These calculations can be performed by a regular general purpose Central Processing Unit (CPU) of the kind you will find in any computer, by a slightly more specialized Graphics Processing Unit (GPU), or a highly specialized ASIC chip designed just for performing that type of calculation.
The relative merits of each method will depend on the hashing algorithm used by the digital currency in question.
Usually CPU mining is the least effective, ASIC mining is the most efficacious, and GPU mining is somewhere in the middle. This is considered to be problematic by some, because the more specialist and expensive the hardware needed for profitable mining becomes, the fewer people get involved in it, leading to mining centralization and therefore a less secure network.
For many people the ideal situation would be for anybody to be able to join in the process of mining on their home computer and still have a chance of earning block rewards. This increases decentralization and therefore creates a more secure and equitable network.
Because of this some cryptocurrencies have been deliberately designed to be ASIC-resistant, meaning that it is difficult to gain an advantage in performing the necessary calculations by developing specialist custom processors. Other coins have even sought to minimize or remove any advantage to be gained by the use of high-end GPU chips, which are not fitted as standard to most computers. This latter type are known as CPU mined coins.