One of the easiest things to forget when traveling to another country is that your chargers & electronics might not be compatible with the outlets at your destination. Let's start with the difference between the two. An adapter is only going to enable you to plug your electronic device/charger into the outlet, while a converter changes the voltage and makes sure you don't ruin your device. You'll always need a an adapter with a converter, but you won't always need a converter with an adapter.
Before you run out and buy anything, you need to check two things:
1. (Converter) The voltage of your destination. Most newer devices have dual-voltage which may include the voltage of your destination, however you still need to double check. This website has a terrific amount of information.
My hair straightener has dual-voltage of 110-230V which means that I can use it worry-free in any country with voltage between 110V and 230V without a converter.
My Samsung cell phone charger has dual-voltage of 100-240V.
One thing to keep in mind, if you are going somewhere that has a lesser voltage than the range on your device you'll need a converter. That means that if I take my hair straightener to somewhere like Japan, whose voltage is 100V, I'll need a converter, however my cell phone charger is fine.
I have a converter similar to this which includes a USB port for all my main chargers (cell, tablet, etc).
2. (Adapter) Check the Plug/Socket type. Now that you have the voltage figured out, you'll want to check the plug size. Details here.
Sticking with the Japan scenario, their sockets are A/B, which is the same as the U.S. so I don't need an adapter. If I go to Chile however, which has C/L sockets I'll definitely need one.
I have an adapter similar to this because they're so compact and you can choose only the one you need instead of a bulky one.
Taking the time to check your electronics will save you from a HUGE headache, instead of getting to your destination, blowing up your cell charger and trying to find a new one in a country whose language you don't speak!