Some programs that run in the system tray provide an option for hiding the tray icon, while allowing the program to run in the background. Three common examples are:
System Clock - Start / Settings / Taskbar & Start Menu, clear the Show Clock box and press OK.
Network connection - Right-click My Network Places and press Properties. Right-click the connection and press Properties. Clear the Show icon in taskbar when connected box and press OK.
Volume control - Control Panel / Sounds and Multimedia / Sounds tab. Clear the Show volume control on the taskbar and press OK.
If a program does not have a hide option, the only way to remove it is to exit the program, kill it, or don't start it. If you right-click the tray icon, you may find a close or exit option. You can use Task Manager to kill the program on the process tab.
To prevent the program from starting, remove it from your Startup folder, the All Users Startup folder, or the HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Run run key.
NOTE: See tip 4221 and link.