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Often times, the answer is via botnet.
A botnet is nothing more than a small program that people install (usually without realizing it) that "phones home" to another computer and carries out whatever instructions it's given.
On it's own, a single infected computer isn't much to write home about. But when you're talking about 100 or 1,000 (and often times even more) computers, a hacker has a lot of computing power at his disposal.
One of the most popular means of gaining access to a computer is to hide a Trojan in an email attachment that joins the botnet upon installation. This happens without the user's knowledge or consent. From here, the botnet owner can deploy commands to his collection of compromised computers.