Computers are broken into by a variety of ways. To name some of the most common methods:

1. The services running on your machine. Services running can open ports which makes you vulnerable to being attacked.
2. Viruses can infect your machine and cause malicious damage. And Trojans (usually destributed by email) can actually open connections to a hackers computer, or open ports allowing a hacker to compromise your PC.
3. hackers can also exploit buggy vulnerabilities in OS's or software. Thats it why it is important to obtain the lastest bugfixes and security patches for your software and OS's.

A Personal Firewall and some good AntiVirus software is highly recommended for the adequate protection for the average home user.

Hope that this sheds a little light.