Basically, this error appears if a file type is set up to use DDE (an old Windows-3.x-style communication protocol between applications running on the same computer) to open a file.
You will find it in file types in folder options of explorer (maybe somewhere else on Windows 7, but there will be some place where an administrator can set up which files to open with which programs).
When that "Use DDE" option is checked, Windows will first check whether the program is already running, and then (if it does) send it a DDE message (consisting of 3 texts, "Application", "Topic" and "Message"). This way it can avoid opening the program again even if it is already running. When the program is not running, it is just started normally. In case the DDE connection fails, the dialog you mention will appear and it will just start the program again normally.
So, the easiest fix is just to disable "Use DDE" for the file extension of your database (I guess .mdb). The only drawback of this: When you open a database and Access is already running, Windows will open a second instance of Access, which will in turn notice the first one and send the DDE message and exit after that. I. e. you are unneccessarily starting Access so that it can close again at once. But nowadays with fast CPUs and hard disks, I guess this is acceptable :-)
The more complex fix would be to check whether something is wrong in these settings (like wrong topic or wrong application) and fix that. Doing that would require you having access to another (maybe virtual) machine where opening files work so that you can compare the DDE configuration.
EDIT: According to this website, that tab was dropped in Vista. You could always do it manually in the registry: Look up
HKEY_CLASSES_ROOT\.mdb for its default value (say it is
mdbfile) and then look at that default value (