What is the factory pattern, and how does a contract deploy another contract?

A factory is a contract that creates and manages other contracts. In Solidity you deploy a new contract instance from within a contract using the new keyword.

import "./SimpleStorage.sol";

contract StorageFactory {
    SimpleStorage[] public simpleStorageArray;

    function createSimpleStorage() public {
        SimpleStorage ss = new SimpleStorage();   // deploys a fresh contract
        simpleStorageArray.push(ss);
    }
}

Contracts can deploy contracts — this is a big part of Solidity's power (contracts freely interacting is called composability). The factory imports the child's code (so it knows what to deploy), and new SimpleStorage() deploys a brand-new instance, returning a reference you can store and later call functions on. A factory typically keeps an array of everything it has deployed so it can act as a manager over all of them.

Go deeper:

• Solidity docs: creating contracts via new — deploying one contract from inside another.