An ISBN carries its own error detection. The final character is 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 consulting any database.
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 because the two formats use different sums. The check digit calculator completes an identifier you have already been assigned, working out its final character.
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 a correct checksum and an assignment are separate things. Identifiers are allocated by the registration agency for your country. No authority sets aside a range for testing, which is why this tool will not invent them: any it produced would fall inside space allocated to real publishers.
No. It validates, converts and completes identifiers you already hold. Issuing them is done by national registration agencies, and no tool outside that system can assign one. Amazon KDP also offers a free ISBN for a paperback or hardcover, with the imprint recorded as Independently published.
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.