With dedicated servers there is a GREAT deal of difference in what different companies call "managed". To some managed is as described above. Basically the initial configuration. The next step is companies that will keep items they install updated for you (OS Kernel, Control Panel Software, PHP, Zend, Perl, Apache, Exim and other such common web server software, but only that which they install and say they support
Then there are companies with various levels of "managed" you can pay for.
Very basic like reboots etc. is free. $X a mo. for security updates & such, $XX for all the things like I described above and maybe $XXX for helping troubleshoot & repair problems associated with things like I listed like when you install an upgraded PHP version and fail to notice it required a Zend update also and don't know why your server isn't functioning properly.
Some companies also include varying degrees of monitoring at various price levels with automatic reboots if a system goes down or automatic repair of hardware. With others monitoring is your business and though they guarantee 3 hour hardware replacement for example that is 4 hours from when you tell them the hardware failed, not 4 hours from the time of failure.
Then there is what I call the "ultimate" where when a company says fully managed that's what they mean and they will help with almost anything you can't handle on software they support (not some 3rd party program they don't offer that you installed) help with all upgrades like apache MySQL. PHP etc. if you need it, even help troubleshoot & fix mail problems or other issues. These are often the companies with the most expensive servers in "base price" and at first you wonder "why are they $50 higher than the same server I see everywhere else?"
This is an area where you must READ & READ & READ to see what a particular company considers managed...then if they offer paid management compare the cost of the level you need plus the server to other companies total cost. LOT's of research & shopping unless you are a 100% capable server administrator and know for sure you will need nothing more than a remote reboot port and you can handle the rest.
Every company it seems has minor differences (sometimes major) in what they do and what is or isn't included. Some also have a "per incident charge" so the management might look cheaper at first but won't be if you have more than 3 or 4 "incidents" in a month.
If you're not 100% confident in your server admin skills how happy you will be will depend on having the right management, the right response times (this varies a lot too, some companies may take minutes, other hours, read the SLA and TOS carefully.)
Then you might also find you could get an unmanaged server and hire a "sever management company" for less in total than a "gold level fully managed" package if the price of "gold level" or whatever is high.
There are so many choices it's unbelievable, and that must be added to your considerations of the quality of the company's datacenters, bandwidth providers, etc. etc.
Take your time and shop carefully.
There was a thread here elsewhere the other day where it was obvious a person had got a totally unmanaged server and had no idea of even how to get the basics installed & up & running. He probably thought "wow, I can get my own server for less than a high-end VPS" and didn't realize what was involved.
Best of luck to you.