Take the calculator's own default, 255 in decimal — a genuinely significant number in computing, since it's the largest value a single 8-bit byte can hold. In binary, it's 11111111 (eight 1s — every bit in the byte switched on). In hexadecimal, it's FF. In octal, it's 377. All four represent the identical quantity; only the counting base changes, exactly the same way "twelve," "12," and "XII" are three ways of writing the same number in English, decimal, and Roman numerals.
Hexadecimal isn't an arbitrary alternative to binary — it's specifically useful because each hex digit maps to exactly 4 binary bits (a "nibble"), with no remainder. 255's binary form, 11111111, splits cleanly into two nibbles: 1111 and 1111, each of which is F in hex — giving FF directly, digit by digit, without needing to convert the whole 8-bit number at once. Octal has the same relationship with binary in groups of 3 bits instead of 4, which is why hex and octal, unlike decimal, can be converted to and from binary just by translating small fixed-size chunks rather than doing full positional-notation arithmetic.
They're all ways of representing the same number using different counting bases — binary uses base 2, octal base 8, decimal base 10, and hexadecimal base 16.
It's a compact way to represent binary data — each hex digit maps to exactly 4 binary bits, making it much shorter to read and write than long strings of 1s and 0s.
This tool works with positive whole numbers. Negative numbers in binary typically use a different representation depending on context.