Most of you might be aware that there are generally two voltage levels followed by different countries. For e.g. 220v standard is followed by countries like India, UK etc. while 110v standard is followed by countries like US. The following article is relevant to you if you have bought electronic/electrical stuff from a country which follows one standard, but want to use it in a country with other standard. And more likely you will be reading this article if you had already bought an item and burnt it by not knowing that you should use a converter or transformer to make it work properly. It may also be the case that you used a converter or transformer but still burnt your device. Don't be ashamed, we both belong to the same community and there are many others.
Before telling if you should use a transformer or a converter, let me tell you that your device may work without any converter. Recent gadgets which are typically carried across the globe by the travellers are compatible with full range of voltage from 100v to 240v. They include mobiles, cameras, laptops, electric razors etc. But check the input voltage rating nonetheless. It should be printed something like "Input: 100-240v". The output voltage rating does not matter.
Now coming to the main point...You should be careful in choosing a converter or a transformer so that your device does not get fried. Most of you might be knowing from your electrical class that you will need a step-up transformer to increase the voltage and a step-down transformer to decrease the voltage. The output voltage of such a transformer should be compatible with the input voltage of your device. Having said that, you should know that there is a different type of devices called "converters" which also does the job of increasing/decreasing the voltage. "converter" is a loose word. They are more accurately called solid-state converters. They are also called SMPS converters. As opposed to the coil mechanism in transformers, converters use electronic circuits to change the voltage levels. The difference in the operation of transformers/converters may be suitable or not suitable for the device that you are using. A quick way to distinguish is that transformers are generally bulky and heavy, whereas solid-state converts are small and lightweight.
If I can to go into a little bit of technical details here, a transformer-based converter converts the voltage by elongating the sine wave of the AC current but still preserving the sine wave formation. On the other hand a solid-state based converts just chops off the sine wave to reduce the voltage, and hence the output current do not follow a sine wave formation. This kind of current is not compatible with all types of devices. But the pure sine wave form of current is compatible with most of the devices.
To summarize: The general rule of thumb is most of the electronic equipment (gaming consoles, DVD players, speakers, mobiles, etc) needs a transformer based converter. Whereas, electrical appliances like motor-based appliances, irons etc can work with solid-state converters. Transformer-based converters can be used for long operating hours whereas solid-state based converts are for short-term use (typically 1-2hrs). When in doubt, it is safer to use transformer-based converters. Also make sure that the wattage(W) rating is compatible. It is safer if the wattage of converter is atleast 25% more than wattage requirement of your device.
Pointers: link1, link2, link3