Mining is really simple. Just run SHA256 hashes until you get the right number of leading zeros. Since hash results are essentially random numbers it takes a lot of hashing to guess such a non-random number. Here's how much hashing it takes:
https://blockchain.info/charts/hash-rate which is 14,000,000 times a trillion times 60 seconds times 10 minutes or 8,400,000,000,000,000,000,000 hashes to get the right answer. Obviously to do that you need fast hashing hardware and a lot of parallelism. The miners have both. An ordinary computer has neither.