A checksum is like a signature of data given in a standard fashion. This representation has a fixed length, and is unique for different strings of data. Files like this usually ends with md5, md5sum, sha, checksum, sum, or something similar.
This feature can be used to verify file downloads. If the owner/maintainer/provider of a file creates a checksum, eg an MD5 or SHA file. You can download the file and also download this checksum file. And once the downloads are finished, you can natively calculate your own checksum using the checksum program. These downloaded and calculated values can be verified, thus telling you that the file downloaded is an exact copy of the original. Another use is to verify larger automatic backup solutions, such as common mirroring applications as rsync or wget.
This system has been used almost since the dawn of computers. And it is used even today without you knowing it, for example there is CRC checking as you transfer TCP/IP packages to view this page.