Yes, the wider connector is the power connector, looks like this:
As you can see, has 4 wires going in, pretty much the same as the older 4-pin Molex connectors. As to why there are 15 connections to the drive, most likely every other connection is a ground. That's done to minimize noise getting thru from the power supply.
If your system's power supply doesn't have any SATA power connectors, then you can buy an adapter that will adapt the older style to the new.
You won't know if you need an adapter until after you crack open your case and look so better get to it!