A file hash or checksum is a semi-unique value generated after running an algorithm against a file. This same algorithm can be used on the file later on to verify it's contents have not changed. If the checksum or hash value is different, it is possible the hosting server has been hacked or that a man-in-the-middle attack has occurred on the network. Some websites offer the checksum prior to downloading so that users van verify the file's integrity after downloading.
File verification is the process of using an algorithm for verifying the integrity of a computer file, usually by checksum. This can be done by comparing two files bit-by-bit, but requires two copies of the same file, and may miss systematic corruptions which might occur to both files. A more popular approach is to generate a hash of the copied file and comparing that to the hash of the original file.