First off, when it did eventually start, did it turn over normally or did it sound like the battery was drained and was cranking really slowly? If it was normal, then it's more than likely something in your starter circuit. You should definitely have the battery checked first, as this could save you a lot of troubleshooting and work. If the battery is fine, then you will want to perform the following tests to try and track down the problem.
First and foremost, check to make sure your battery cables are clean and tight. Disconnect them and clean the terminals and battery posts with a wire brush to make sure you get a good connection. Reattach them and make sure they are tight.
If this doesn't change things, check the starter relay (the thing in the picture). You see the two heavy wires on the left side? Those are your cables coming in from the battery and going to the starter. You should be able to jump those terminals with a large screwdriver or a pair of pliers (wear goggles because it will most likely spark a bit). If the bike cranks over, it's most likely the starter relay, the ignition switch on the bars, or the wiring in between. If not, you probably have a problem with your starter or the cable going down to the starter. (P.S. If anybody knows this to be the wrong way to test the relay, please speak up). By the fact that you are hearing the click coming from the relay, I would think the ignition switch is fine.
If it still doesn't turn over, you will want to remove those cables and clean them the same as you did with the battery cables. Make sure they are tightened good as well. And be sure to disconnect the battery before cleaning them. Clean the cable that connects to the side of the starter in the same manner and make sure it's tight as well. If this doesn't make any difference, you most likely have a bad starter.
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06' FLHRSI Road King Custom (Buzz)