Firstly are you sure its capacitor, not a diode or resistor?
Do you know the value/specs of the capacitor you removed?
If you don't, can't read the value (on the component itself) or can't find it, then I would highly suggest you call it quits.
There are hundreds of thousands of different capacitors each with different ratings, form factors, tolerances, etc.
If you know the value/specs of the capacitor do you have a supplier?
On the basis you have the correct replacement capacitor you'll need to find out if it was soldered through hole or surface mounted.
Easiest way to find this out is look on the bottom of the motherboard and see if there are legs directly under the capacitor you removed. If there is its through hole.
Either way you will need flux, solder and either a solder sucker or/and de-soldering braid (both highly recommended).
Yes it costs money but without flux its going to be hard to solder and you risk the solder running onto other parts. You need solder to solder the new capacitor and you need the solder sucker and/or braid to remove the old solder. If you don't have it then the solder will be much much harder to remove and you run the risk of it running onto other parts.
I find the easiest way to de-solder through hole is to hold the pcb in a vice, use the soldering iron to heat one side then use a solder sucker on the other side. That way the colder solder is sucked up first. If you do it from the same side you heated the cold solder is the last to be sucked up meaning more attempts.
De-soldering smd (surface mounted) is easier using de-soldering braid, put the on the solder, apply the soldering iron to the braid. Remove when the braid is silver or coated in solder.
Either way its going to cost you atleast $10 to fix if not more and thats if you can find the part.