This is very standard question, which I come across from many new users. Generally there is some confusion prevailing between two ESRI technologies MapObjects and ArcObjects.
MapObjects and ArcObjects are two completely different technologies. MapObjects is very limited in what you can do. If you write an application in MapObjects and intend to distribute it outside of your organization, then you are required to pay ESRI a royalty fee for each installation of the application.
Applications written in ArcObjects under ArcGIS 8.x do require the user to have ArcGIS installed on their computer. ArcObjects, starting at release 9.0, can be used to develop stand alone applications through the ArcGIS Engine. The Engine licenses will offer varying degrees of functionality and will not require your users to have ArcGIS installed on their computers. Your users will be required to purchase an Engine license, which will be cheaper than purchasing an ArcGIS license (but not as cheap as the MapObjects royalty).
However, if your application is simple, MapObjects may be the way to go. ArcObjects is much more powerful. It may be that your app would require ArcObjects in order to provide all of the functionality.