PGP is useful for two things: 1. Privacy and Security, and 2. Authenticity. By privacy, I mean that you can prevent people from seeing things. For example, you can encrypt an email to someone, or encrypt a file with a list of passwords. By Authenticity, I mean that you can ensure a message was sent/written by the person you think it was, and that it wasn't modified by a third party. Of course, these two can be combined. I'll discuss these further.
By using PGP for encryption, you can send something private or sensitive, or have someone send you something private or sensitive and be sure no one else can read it - not even the administrator of your email server. I should note that for this to be effective, unencrypted copies should not be kept, obviously.
Group passwords, account numbers, etc. can all be passed securely between parties that need to convey such sensitive information.
But it doesn't just help for email. GPG Encryption gives you a way to store private data for yourself securely. You can have an encrypted file with all of your online passwords, or your bank account information, or anything else you might want safe.
Another examples is encrypting data before you send it off to online storage so no one can read it.
Regardless of what you want to keep safe, PGP provides a free and safe way to do it.
If you need someone to be sure you really sent something, you can digitally sign an email, a piece of software, source code, or anything else.
By doing this, anyone reading the email, or downloading that software can be certain that:
This is a secure way for managers to provide directives to their employees or for students to ensure communication from their University isn't forged. It allows software users and authors to ensure that trojans weren't slipped into programs unexpectedly by a hacker.
But some people take it further than just signing important emails - they sign all of them. Why?
Well if I sign all of my emails as a matter of policy, than no one can forge email as me. Any email that comes from me but isn't signed is clearly a forgery! And when it's as easy as setting your email address to firstname.lastname@example.org in your mail client and sending off an email, why would you not want to protect yourself from this?
What if all communication from Universities, ISPs, and corporations were signed? No more Ebay and bank phishing scams, no more viruses from email@example.com, no more "this is support, please respond with your password." Why? Because if every piece of official mail from major institutions was signed using their private key, then people could easily ignore any email that doesn't validate. The large red warning in mail clients would drastically cut down on people clicking on bad things (although rest assured, some people would still do it).
By the same token, if you sign all of your email, people will know that the virus didn't came from you when it's not signed and that the prank email to the entire office wasn't from you.
Of course, not everyone needs to, or can, sign every email. But the mechanism is still there for signing things that need extra authenticity and verification.