Generally speaking if the battery isn't charging, you've either got a defective battery or need a new alternator or stator. Since you've got a Ski Ray in your photo, I'll assume you don't have an outboard so you'd be dealing with an alternator instead of a stator.
Go to Radio Shack and pick yourself up a digital multimeter (get the cheap $15 one) and check the voltage coming out of the alternator at idle. if it's lower than 12V or higher than 15V, then your alternator is the issue. A more reliable test would be using an ammeter but the amps coming out would vary based on the alternator. You'd have to check a service manual to see how many amps should be coming out.
However, while it's not common, you may have bought a defective battery. Use that same multimeter to check your voltage at the battery terminals. You should be getting a similar reading.
If that doesn't work, I'm out of ideas other than hunting for shorts but I'll bet money it's your alternator since your battery is new enough to not be dead but old enough to probably not be defective.
Alternators are pretty easy to replace but are sometimes hard to get at. Make sure to get a marine alternator if you need a replacement.