Phishing is a more serious attempt to get hold of personal and financial detail. Typically an email or web page will direct the user to a site which looks like a bank site. The user will then be asked for bank details - passwords, accounts etc. Because the site looks genuine, many users have been known to type in these details. The fraudsters then use the details to withdraw money from the account.
No bank, building society or financial institution will ask for security details to be sent via email, nor will they ask that they be re-typed into a web page. The solution is not to give away details.
Many anti-virus packages evaluate the safety of web pages and Internet explorer has anti-phishing built into it - make sure this is turned on.