What does it mean to "subscribe" to an IMAP folder?

"The folders you subscribe to are the ones displayed by Thunderbird. It subscribes every folder it finds when you first setup the account and any folder you add through Thunderbird is automatically subscribed as well. I guess that option is there so if you were to add a folder through another application or a webmail-ish app and you wanted that new folder to be displayed as well you can just turn it on.

Maybe this link will clarify things. Here's a quote:

Your IMAP email client (eg
Thunderbird) will display the folders
in your account to which you are
""subscribed"".

Generally, this will be all of the
folders in your account, and only the
folders in your account. However, in
some situations you may find that
there are folders in your account that
do not appear in Thunderbird, or
folders that appear in Thunderbird
that are not in your account. This can
be fixed by subscribing to (or
unsubscribing from) the offending
folders."

"There are already a couple of very good answers to this question (which I’ve upvoted) but I thought I’d complement them with an answer detailing the relevant commands specified by the IMAP protocol, RFC 3501.

Subscribing to folders

In IMAP, subscriptions are used as a way of marking which folders should be displayed by IMAP clients. Some system folders may provide functionality but not contain emails that are of interest to the user, e.g., an Exchange server contains folders such as Contacts, Journal, Sync Issues).

Two IMAP commands are used for listing IMAP folders:

the LIST command lists all folders.
the LSUB command lists only subscribed folders.

The output of the LSUB command is affected by 2 commands:

SUBSCRIBE adds folders to the list displayed by the LSUB command.
UNSUBSCRIBE removes folders from the list displayed by the LSUB command.

See What is the use of Subscribe and Unsubscribe commands in IMAP4? for examples of the effects of the subscription commands.

Note also that folders are marked as subscribed on the IMAP server so that all clients see the same list of folders. See Is IMAP (un)subscribe meant to work across mail clients?."