I am not just saying this because I am a lawyer. You need to see a lawyer. The media lawyers are threatening all kinds of things, throwing all kinds of case law at you. Above all, they are trying to get you to settle before you figure out what you can do to defend yourself. If you find a lawyer who is into this, the lawyer will likely give you a free consultation, read the threat letter over, and tell you where you really stand.
Save your paper trail
Many people have access to your ISP address. You need to document your alibi. Were you even there when the alleged download occurred? Can you prove you were out? Is it possible someone else could have done the downloading? You need to preserve your papertrail that can lead you out of this mess.
Reach out to other defendants
The copyright holders are problably going after a bunch of you. But in the beginning of the lawsuits, the defendants are listed anonymously, as "John Does." The anonymity leaves you isolated and vulnerable to the plaintiff. They will try to pick off settlements quickly before you can connect with each other. Because they know if you get together, you can all mount a joint defense that will beat them back. So see if you can find other defendants.
Be prepared to settle
Copyright holders are routinely settling for about $3,000 against individual, amateur downloaders. If the same amateurs take it all the way to a judge, many judges are accepting $750 to $1,500 settlements as reasonable. Before they can get there, though, the copyright holder is going to have to prove it was actually you who did it, and face discovery demands that you can throw at him.
Don't do it again
Let's face it. We need to support artists in our world. It is not fair to rob them of their livelihood, just because you can. By doing it, you're helping diminish art in our world. If you escape the copyright holder's clutches, let it be a lesson to you, and start paying for the art that you enjoy.
Additional resources provided by the author
The Electronic Frontier Foundation has good advice for defendants at its website, eff.org.
Torrentfreak.com is a downloader-friendly website that keeps track of these lawsuits around the world.