Ok, so lets say you download 64-bit Linux Mint. You then download the checksums file:
Many distros have separate checksum files that are specific to the iso that you download, while Mint has all the checksums in one file.
So after you download the linuxmint-16-kde-dvd-64bit.iso and the sha256sum.txt into the same directory, then you can open a terminal at that location, and run 'rhash -c sha256sum.txt' and that will check all the checksums listed in the file against the iso if it exists. You can also do the same using 'sha256sum -c sha256sum.txt', and most checksum programs use the '-c' option to check checksums.
Another way you could do it is just run 'rhash linuxmint-16-kde-dvd-64bit.iso' and check the checksum yourself, visually with the listed one.
If this is not the answer you wanted, then please provide a concrete example of what is needed.