The Ethereum frontier network has a customised proof of work algorithm called Ethash. What kind of computer hardware is required to mine with this proof of work algorithm?
You can mine ether using your computers graphics card(s) (GPU) as long as it has enough memory to hold and access the DAG efficiently. To be efficient and competitive you should select a card with high total memory bandwidth. You could technically also mine with your CPU, but the current difficulty on the main net is so high that you wouldn't mine ether in years.
You can mine ether in the test net with your CPU, as the difficulty is not high there, but they aren't having any value, besides using them for testing.
As you pointed put Ethereum uses its own Ethash algorithm to mine. This algorithm is made to be memory hard to make it hard to develop ASICs for it. Its generates a 1GB+ DAG file, which needs to be replaced once in a while to mine.
You will need at least a 2GB RAM on your GPU to actually mine. CPU will be extremely hard to make any money unless you use a pool. Here's some resources: