Phishing is a way of attempting to acquire information such as usernames, passwords and even credit card details by posing as a trustworthy company in an email. The most common versions of this kind of email are emails which look like they have come from banks or social media sites saying there is an alert and you need to log in to deal with it - however, if you click on the links they will take you to fake sites where they capture your information.
This technique is not just about the look and feel of emails but also the URLs behind links. In some emails where you would expect the link to take you to a website, such as www.natwest.com, the actual link behind it could be completely different and will send you to places you definitely don't want to go!
As with everything online, please double check that you know the sender and you do everything to protect yourself online!