When you decide to delete an email, it is sent to the trash folder where you can still access it if needed but only for 30 days after which it may automatically be removed from the system and unrecoverable.
Archiving an email in Gmail is taking that email and placing it in a container called All Mail where it will continue to exist but will no longer be cluttering your inbox. Archiving will allow you to locate that email using Google's powerful search features.
Why should you archive your emails? Say you're having a conversation with someone and information like an address or phone number is exchanged. At that moment you are able to write down or save anything needed so you decide to delete the email. Now imagine it's months in the future, you lost that information and you want to retrieve it again. If you deleted the email you're out of luck. If you archived the email you will no longer see it in your inbox but a quick search will locate it.
How to Archive:
Gmail has two options in the top menu bar for dealing with unwanted email. Click the checkbox next one or more messages in your inbox list then click the Archive button. The trash can button is for deleting.