If the cap is covered with a plastic sleeve it is an electrolytic cap and IS important.
The markings show it to be a 10 uf (microfarad) 10volt size. This is usually used close to an IC (integrated circuit) as a DC power filter. They are all over the board, some larger, some smaller.
Even though the PSU outputs well filtered DC, as the power is distributed all over the board, spurious electrical noise is created by fans, switches, motors on the hard drives, etc.
You are very wise to be hesitant to put a soldering against the circuit board if you have not had considerable experience in do this. One reason is that most circuit boards in computers are not just double sided. Most are layered meaning that there are traces for circuits sandwiched between the inside layers of the board.
If you look closely you will see what appears to be tiny holes through the board. These are called "vias" which just means that contact is made to several circuit points on both sides of the board and through the internal layers using the "vias". The holes for the capacitor leads MAY also use vias so the leads contact more than one point in the circuit.
These are very delicate and sensitive to temperature limitations form soldering irons. In addition, the contacts are so small that is takes a very fine point temperature controlled soldering iron to safely work on them.
With this situation and the fact that you are not comfortable yet working on a board to this level, I strongly advise you to take the capacitor and the board to a certified repair shop.
