Gmail was the first email provider to encourage its users to store, rather than delete, messages. Gmail accounts have an increasing amount of storage to cope with this ever growing stack of emails. Archived emails are found under the All Mail label (accessed by swiping in from the left of the screen and scrolling down).
All messages already have the All Mail label and the act of archiving messages just removes the Inbox label, thereby removing them from the inbox. Archiving messages can be done in two ways. Firstly, from the inbox view, just swipe an email to the right. If you are viewing an email, it can be archived by tapping the central icon on the top row of the message view (which resembles a box with a downwards arrow).