"Phishing" is the term coined by hackers for attempting to lure personal information out of people by pursuading them to visit web sites that look like genuine bank, credit card, or payment sites, when they are actually sophisticated fakes of those sites.
These attempts at fraud appear as email messages sent to you asking for some sort of personal information. Usually this is information about your address or credit card information, or your cash card PIN number. It could also involve you typing in some sort of password to "gain access" to a security site where they want you to type in more information.
You should treat any email with suspicion when it takes you to a site asking for this information. If it is from your credit card company, why do they need your credit card number and expiration date? Don't then already have this? Likewise, why would your bank need your PIN number? They already know it. Just as if someone walked up to you in the street and asked you for your PIN number, treat emails that are apparently from your bank with the same care.
To be safe. Don't click on any links in email messages purporting to be from your bank. Type them in by hand, just as you would when normally doing banking over the internet. This ensures you go to the site you typed in and not some site pretending to be your bank. Walking down the street, would you happily walk into a new shop which happens to have your bank name printed over the door, and give the staff your PIN number?
Internet banking is still very safe. But be careful, don't give your personal information to anyone who asks, even if they do look like your bank.
For more information on malicious emails, visit out Notes About Malicious Emails page.