From text to bytes
Whatever you type is first converted to UTF-8 bytes — a real difference from "characters": an emoji or accented letter can take up several bytes, and SHA-256 hashes those bytes, not an abstract idea of "characters".
Padding to a whole number of blocks
SHA-256 only processes complete 512-bit (64-byte) blocks. Your message is padded: a single 1 bit is appended, then as many 0 bits as needed, then the original bit-length as a 64-bit number — leaving a result that's an exact multiple of 512 bits. For a short string like this tool's defaults, that's a single block.
64 rounds of mixing
That padded block runs through 64 rounds of bitwise rotation, shifting, and modular addition, updating eight 32-bit working variables derived from SHA-256's fixed initial constants. The result becomes the chaining value fed to the next block — the Merkle-Damgård construction covered in the Hash functions module — or, since there's only one block here, the digest itself.
Why editing one character changes everything
Each of those 64 rounds is designed to spread every input bit's influence across the entire internal state within a handful of rounds, the same avalanche principle behind AES. Change a single UTF-8 byte at the very start, and by round 64 essentially every one of the 256 output bits has an even chance of flipping — exactly what you see comparing the two digests above.