You have two choices:
a) Powered HDMI switch
http://www.monoprice.com/Product?c_id=101&cp_id=10110&cs_id=1011001&p_id=2786&seq=1&format=2 , or
b) Powered DVI switch
http://www.monoprice.com/Product?c_id=101&cp_id=10110&cs_id=1011004&p_id=4069&seq=1&format=2
The mechanical version is cheaper and tends to have better compatibility, however I don't see the point in doing it that way since it's not much different than getting up to switch the monitor input.
*Some monitors will AUTOMATICALLY switch inputs if you turn off a device and another one is present or turned on. My dad's old Sony does, my newer Dell does not.
You'll also need the cables. I recommend an HDMI switch, and use the HDMI output from your PC if you have that or go with a DVI-> HDMI adapter or cable. Recommend the cable.
You'd need an HDMI->DVI cable or HDMI->DVI adapter + DVI cable to go to the monitor.
Other:
*Make sure your XBOX 360 works with the monitor, and that you have an AUDIO solution enabled. Without HDMI on the monitor you'd need to use whatever separate audio output the console has so that may be an issue.
Solutions:
http://support.xbox.com/en-CA/xbox-360/connections/connect-sound-system
and
http://support.xbox.com/en-US/xbox-360/system/connect-xbox-360-console-tv
*If you are switching the VIDEO input you also need a way to switch the AUDIO input. Unfortunately the ideal way is to have a monitor that supports HDMI, otherwise you may need to split the AUDIO input into desktop speakers which may cause issues with audio strength or noise by having multiple devices attached.
TOSLINK (digital) would be a better way if you have speakers that support this and perhaps have the XBOX on that, and the PC on the analog.
Summary:
Must think this out carefully for both VIDEO and AUDIO. If your monitor is inexpensive, perhaps get a replacement with dual HDMI, audio pass through (RCA), speakers, and ensure it can auto-switch between devices when one is turned off.
Example with new monitor:
BOTH the PC and the XBOX be attached via HDMI (2xHDMI inputs), and have the speakers attached to the likely RCA output (may need a simple RCA->3.5mm adapter). DisplayPort may work with audio in some cases but it must be supported on both ends (sound card output, and monitor input).
Further, the monitor would auto-switch if one of the devices was turned off.
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