There are many flight comparison websites out there. Here are some of the best:

Often you will find the best prices by using these sites to scan through many different airline fares.

Google Flights is a fantastic, flexible way to search for flights. It shows you the best features of a city destination (such as concerts, nightlife, shopping, etc), it gives you ideas if you are not sure where to go, you can set parameters such as your date range, price range and so on.

Often, this should be your first stop when travel planning.

Matrix Airfare Search is for more advanced users. ITA Software, the company originally behind Maxrix Airfare Search, is the 'secret sauce' behind Google flights after being bought by Google, think of it as a more advanced way to search.

Kayak is a wonderfully intuitive way to search for flights. You get graphs of price over time, choice of airlines, which you can tick on or off, you can pick your preferred airline alliance (e.g. OneWorld or Star Alliance). Highly recommended.

Sky Scanner is another popular website, if you need another option.