What exactly does it mean to "archive" an e-mail in GMail?

It goes into the All Mail folder. Your All Mail folder contains all the emails you send and receive, that you’ve saved. Your trashed emails won’t be in it. When you archive an email, it goes into your All Mail folder, and it is saved so you can go back and retrieve it. If you want to stop a chain of mail because your conversation was too long, you can archive it, and when the person replies to the last email you sent before you archived, it will be a new chain.
I don’t archive mail, because I like to be organized, so I use folders.

Your All Mail Folder contains pretty much any email in your account thats not in the trash. Inbox, Sent, Drafts, everything’s tossed in there like a big box of junk. Gmail doesn’t use a folder structure like other mail services do, it’s run by Tags, and by there is how you organize it.

For example, a car insurance bill comes in. I can tag it with something like [Car], or [Bills] and then just send it to the archive folder. Then when I wanna pull it up again, I type in the search box for either the name of the company or just the tag names and it will bring them up. also, a list of all your tags is on the left of the screen so it “feels” folder like as well.

Instead of deleting the email, it’s simply saved in an all-purpose folder to be retrieved later, if needed. When you want to find an email that you’ve archived, just search for it in the search bar. To make them easier to find, you can give emails tags.

Everyone above is right, but this is the way I think of it-

There are several folders in your email: The Inbox, Spam and All mail being only three

However, these each serve different purposes:

The Inbox
This is where you receive emails. This is a temporary location. It is much like a mailbox, it is not a storage place for mail, just a receiving place.
From here you can do two things, you can hit spam or archive

Spam
This is for all the ads and unwanted mail you receive.

All Mail
Now this is the final resting place for “good” mail. Here you can keep a record of every email you receive, not the inbox.

So thats the way I feel about this. I always strive to have no messages in my inbox

I had archived several chats and went to look for them and could not find them so I don’t know if chats are archived differently from mail. When I restored the person to my chat list, I could retrieve the chats but not before.

Technically, Gmail’s inbox isn’t an inbox like it is in all/most other emails – that is, it isn’t all of the mail that you have received. Rather, your inbox is a tag that all incoming mail gets until you remove it (unless it’s spam, or you have a filter telling Gmail to not tag it as inbox). All Mail contains all your mail, both received and sent, except for those marked as spam or trash. When you “archive” an email, you’re simply saying that you no longer want the inbox tag on it for quick and easy reference. I think of it sorta as if Twitter let marked all unread tweets (a feature it needs…) as #unread, and then let you manually remove the hashtag.