It would be great to have some more details about the application and the resources it uses, but by the error message, I would think that the application opens some file in “write” mode, and that’s the reason why it causes that message to show when you try to run another instance.
Usually when a file is created, the creation instruction specifies (among other things) how the object can be shared (or specifies that it cannot be shared at all), and which action to take on files that already exist. This sort of settings can cause the problem you are facing if the file being open reside on a network shared resource, and can be accessed from different machines.
More information will definitely be needed here. Applications which talk to SQL Server are designed to be used by more than one person. It sounds like this might be an Excel sheet, or Word Document which pulls its data from a database.
The problem will be with the front end, not the back end database. Can you provide more information about the front end?
An easy (but hard to manage solution) would be to put the front end (I’m assuming Excel here) on each users computer so that each user can open there own copy of the Excel sheet.