Headlight trouble-shooting

Recently my passenger side HID bulb went out, so I bought a pair and installed the new one. Didn't work. I've changed both bulbs in the past successfully, so I don't know what the problem is here. I'm asking for advice about how to find out what could be the problem besides the bulb. A fuse? Relay? What's a relay? I don't know where to start here. The bulb worked fine for about 6 months, then the hi-beam went out, then it went back on for a while, then the whole thing went out completely a couple weeks later. Who knows?!