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"Compressed public keys were introduced to bitcoin to reduce the size of transactions and conserve disk space on nodes that store the bitcoin blockchain database"
For every private key we can generate compressed and uncompressed legacy (starting with 1) bitcoin address*. Every public key contains x and y coordinates of elliptic curve: for uncompressed the public key contains both coordinates, but for compressed only x coordinate and "sign" for y - so the compressed key is less in size, and less in transaction fees as well.
More details are here: https://github.com/bitcoinbook/bitcoinbook/blob/develop/ch04.asciidoc
* Of course there are also segwit (starting with 3) and bech32 (starting with bc) bitcoin address. They are not generates by my tool. However segwit and bech32 also generated from compressed public key.