Whether you delete or archive an email message, it disappears from your inbox. A deleted message goes into the trash folder, but an archived message is defaulted to the Archive folder or All Mail on Gmail / Google Apps.
Depending on your email provider or company's policy, your deleted messages may be permanently purged from your trash folder automatically (often after 30 days). Your Archive folder is never automatically emptied. Any email you see in either the trash or archive folder can be moved back to your inbox folder.