This capacitor is known as the clock cap. It keeps the correct time in the xbox dash/es by feeding it's juice into the clock circuit in the event that the ac power is interrupted.
You can remove it safely ONLY if you have a chip containing a bios with error 16 support (clock fix), or,
if you are not sure what bios your modchip has, as long as you have a working M$ dash present on your C drive, you will be OK.
In the event that your AC is interrupted, the M$ dash is forced to load so the clock can be set. The M$ dash must be present and working.
Bios' with the "clock fix" will simply ignore the invalid clock state and load your hacked dash anyway.
It may be possible to safely remove it in the case of a softmodded xbox, but I am no softmod Guru, so I can't say for sure. If you have a softmod, wait for a reply from someone who knows softmods well. Also, if you do have a softmod, you may want to leave your xbox plugged in to the AC outlet until someone with better softmod knowledge can chime in.
As far as the mess, you must clean that electrolytic fluid (yep, the dark area surrounding the cap is a puddle) off of your board or it will continue to eat the board until catastrophic failure occurs. It is corrosive.