I think the term you are looking for is "hosted database" or "database as a service". Database management service is a bit vague. If you are comfortable around

I think the term you are looking for is "hosted database" or "database as a service". Database management service is a bit vague. If you are comfortable around a Linux command line shell, I suggest getting a VPS (AWS EC2, Linode, etc.) and simply installing MongoDB using your distribution's package manager. But it sounds you are looking for a more managed service and don't really want to bother with system administration (which is perfectly fine). In that case, I would recommend going with a PaaS provider like Heroku which offers a number of out of the box hosted services including (but not limited to) MongoDB.