When you ask someone to email you using your Gmail account, you can ask them to use a plus sign and their initials (or whatever you want) right before the @ symbol in your email address. This does not give you a new email address. It is only an alias. You can use this for inbound emails only.
When you receive an email that has the alias, it goes to your main email account but it goes to its own folder automatically. This way you can better organize your inbound emails.