Without much understanding of electricity, you should be able to isolate your problem. Using one of those "ice pick" style 12 volt testers that you can buy in an auto parts store, you could take the plug off the back of headlight (or the high beam light if it is a 4 bulb setup) and probe the connection to see if 12 volts is present in the high beam position. If not, you are back to the suggestions of the other posters, the switch or the relay. If I were a betting guy, I'd bet on the relay. BTW, when I use the ice pick style tester, I often have use another piece of wire on the ground lead to get me to a good ground source on the chassis. The wires provided on those testers are just too short much of the time.
2000 Georgie Boy Landau 36' DP
2005 Saturn Vue toad