The Bitcoin Hash is calculated by taking the input data of a block of transactions, running it through a hashing algorithm (in this case, SHA-256) which outputs a fixed-size alphanumeric string. This string is then compared to a Target hash. If the output string is less than the Target hash, the block is considered valid and is added to the blockchain.
If it is greater than the Target hash, it is considered invalid and rejected by the network. The difficulty of the puzzle (Target hash) is adjusted so that on average, a new block is added to the blockchain every ten minutes.
The Bitcoin Hash Rate is the number of times the hashing algorithm can be run per second. It is a measure of how fast the network can process transactions.
The higher the hash rate, the more secure the network is from 51% attacks (where someone could potentially gain control of more than half of the computing power on the network and double spend coins). As of writing this, the Bitcoin Hash Rate is approximately 61 EH/s.
To calculate the Bitcoin Hash Rate, simply take the number of hashes per second that your machine can calculate and divide it by 1 million. For example, if your machine can calculate 4 million hashes per second, your hash rate would be 4 MH/s.