Archiving an email removes that email from your inbox and places 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 recall that email using Google's powerful search features and does not delete 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.
Deleting an email by clicking the trash can removes that email from your inbox and places it in a container called Trash where, after 30 days it will be permanently deleted and reduces the storage that your account is using.