What is a library in Solidity, and what does using ... for do?

A library is a contract-like collection of reusable functions that holds no state and cannot receive ether. using Lib for Type attaches the library's functions to a type so you can call them as methods on values of that type.

library PriceConverter {
    function getConversionRate(uint256 ethAmount) internal view returns (uint256) { ... }
}

contract FundMe {
    using PriceConverter for uint256;

    function fund() public payable {
        require(msg.value.getConversionRate() >= MINIMUM_USD, "not enough");
    }
}

Library functions are internal and stateless. using PriceConverter for uint256 makes msg.value.getConversionRate() valid: the value it's called on (msg.value) is automatically passed as the function's first argument. This is syntactic sugar that reads cleanly and keeps math/helper logic out of the main contract. Any extra arguments go in the parentheses as usual.

Go deeper:

• Solidity docs: libraries — stateless reusable code and how it links.
• Solidity docs: using for — attaching library functions to a type as methods.