I'm planning to upgrade to a Radeon HD 4890 but I found in the requirements that I must have a monitor capable of HD.
This is the monitor I have right now:
http://www.newegg.com/Product/Product.aspx?Item=N82E16824236056

How could I know if it's HD capable?
And if not, what do I have to search on a monitor to know if it's HD capable?
That monitor appears to have a maximum (or recommended) resolution of 1680 x 1050. HD is higher than 1280×720. Your monitor is HD capable.