Signing means attaching electronic signatures to mails or even software to prove its correct derivation. To avoid that someone writes mails using your name and to protect both you and the people you send them to, you should sign your mails. Signatures help you to easily check the sender of the mails you receive and to distinguish good-natured mails from malicious.
Software developers sign their software so you are able to check its integrity. Even if you have the software from an unofficial server, you can verify the package with the signature.
You might have sensitive information you want to protect from other parties. Encrypting helps you to transform data and make it unreadable for others. This is important for companies it is important to protect internal information as well as their employees' privacy.