The "round thing" is a starter relay. You have power going from your battery direct to one side of this relay. The wire to the starter is on the other big terminal. There is also a wire that puts voltage to the relay when you push the starter button. This triggers the relay to close and power goes from the battery to the starter. Sounds like your battery may be OK, the relay may be OK but the bike doesn't want to crank.
If the bike sat outside, what have YOU DONE TO IT prior to trying to get it started? Have you changed the oil? I don't know HOW the starter clutch works on your bike but it may be the problem or the bike may not want to turn over because the oil is totally gummed up. If a bike sits for awhile, you are wise to perform full tune up and maintenance procedures before attempting to operate it. If someone besides yourself did these things, you should probably do them as well yourself just to be sure things were done correctly.