Maintenance mode is a way to disable web access to your site while leaving SSH (and FTP, if enabled) access available. It is suitable for use when you need to edit your site, but you don't want people accessing it while you do, for example if you find out you urgently need to apply a security update to a software package you are using for which an exploit is already available, or if you just don't want people accessing the site while you are in the middle of making major changes to it.
If we discover a problem with your site, such as if it is being exploited to send spam due to a vulnerability in software you are running, we will frequently convert your site to maintenance mode to stop the spam outbreak but leave the site accessible to you so you can fix the problem. Once the problem is fixed, you can bring the site back online yourself. (But if we shift your site to maintenance mode, please make sure you solve whatever problem required us to do that before you bring it back online; we have other ways of disabling sites that our members cannot reverse on their own, and we really dislike being forced to use them.)
The message presented when people attempt to access your site is the same one presented if we perform system maintenance that requires us to take your site offline, and can be seen here, so if you need to work on your site discreetly, you do have the option to put it into maintenance mode and blame us.
To tell the difference, check the Site Information page. If it says the site status is "Member Maintenance" then it is under your control. If it says the site status is "System Maintenance" then we are doing something that requires the site to be offline. (Our system is designed to avoid the need to do this in all but a few rare cases.)
To enter or leave Member Maintenance mode, use the relevant link in the "Actions" box on the Site Information page for the site you want to affect. System Maintenance mode is controlled by our system. Your site status will change automatically to this status when such maintenance starts, and change back once the maintenance is complete.
There is a third type of maintenance mode, called "Restricted Maintenance." Most members never encounter this, but when a site has a bad track record of problem behavior (e.g. being hacked to send spam) and the site operator demonstrates unwillingness or inability to resolve the issue, the site will be placed in this mode. There are two ways to get a site out of restricted maintenance: delete it or, if you are a subscription member, open a support issue explaining what you have done to resolve the problem and asking for the site to be reviewed.