A watt is a unit of power that is equal to the voltage of a power source times the amperage:
P = V * A
A lot of times, the amount of power you can get from a solar panel is expressed in watts. This is because most of the time, this is how people will equate it to their power bill. This is fine if you intend on sizing a solar panel system that will be converted to AC power and sent into a home or power grid. Sometimes, such as building a panel to directly charge a battery, or an iPhone, it is more important to pay attention to the voltage and amperage, to make sure that you are getting the correct kind of power, not just a same amount.
Look at your iPhone for instance. The back of your wall charger says that it provides a 5V, 1A current to your phone. That totals 5 watts of power. If you try charging your phone with 10V, .5A source, you may damage your phone. Charge it with a .5V, 10A source and you may start a fire.
The bottom line is, if you intend on using watts to size your solar system, you are probably sizing your system with the intent of using inverters and regulators and other things of the nature. If you are looking to size a solar panel based on what you will need to directly charge an appliance, make sure to check out our Solar Power Supply Informational. If you are trying to size a residential system, make sure to check out our Residential Solar Power Supply Informational.
Before we wrap this up, lets equate this to your power bill. Say you were charging a giant battery with your USB charger. If it took 10 hours, you will have charged 5 watts for 10 hours, meaning you used up 50 watt-hours. That is how your power bill adds up your usage.