Applications in OS X are by default assigned to the current desktop, so if you open a program and then switch to a different desktop, the program's windows will stay on the current desktop. However, there are a couple of ways to change this, so specific programs open on a different desktop, or so they remain present on all desktops.
The first option is to assign a program to all desktops, which is the most straightforward option. To do this, simply add the program to the OS X Dock, then right-click it. In the contextual menu that appears, choose "All Desktops" in the "Assign To" section of the Options submenu. From now on when you open this program, its windows should always be on your screen, regardless of which desktop you switch to. This option may be useful for monitoring utilities such as Activity Monitor or the Finder.
The second option is to assign a program to a specific desktop, or "space." To do this, first invoke Mission Control (this can be done with the preassigned F-key configured on most Mac keyboards), then go to the desktop of choice. Then right-click the application in the Dock and choose "This desktop" from the same contextual menu. Now, regardless of the desktop you are on, if you open or switch to this application, you will be switched to its assigned desktop, unless a its window has been manually moved to the current desktop. This can be useful for keeping programs grouped by type or by function, such as word processing and spreadsheet programs so your office work can be kept sequestered.
Finally, to undo these options for any program, you can right-click it and choose "All" in the "Assign to" contextual menu section. This will again have it be assigned to the active space only for its current session.