An ISBN carries its own error detection. The final character is not part of the identifier so much as a receipt for the digits in front of it, calculated from them by a fixed sum, which is why a mistyped ISBN can be caught by arithmetic alone without any database being consulted.
That is what these tools do. The validator recomputes the check character and tells you whether the number you have is internally consistent. The converter moves an identifier between its ten-digit and thirteen-digit forms, recalculating the check character for the target format because the two use completely different sums. The generator produces correctly structured sample numbers for testing layouts and import routines.
The distinction that matters most: none of this issues you an ISBN. A number can pass every check here and still belong to nobody, because validity and assignment are separate things. Real identifiers are allocated by the registration agency for your country, and the samples this tool produces are labelled as samples for exactly that reason.
No. It validates, converts and produces samples for testing, but issuing identifiers is done by national registration agencies, and no tool outside that system can assign one. The generated numbers are structurally correct so they behave normally in a spreadsheet, and that is the limit of their purpose.
Each digit is multiplied by a fixed weight depending on its position, the results are summed, and the final character is whatever makes that total resolve against a set modulus. Changing any single digit changes the sum, so the number is rejected. The two formats use different weights and moduli.
Its check character is calculated against a modulus of eleven, which produces eleven possible outcomes for a system with only ten digits available. The letter X stands in for the value ten. The thirteen-digit form uses a modulus of ten and never needs a letter.