Once you have received most viruses - if you open the attachment to the message, the computer is affected. Unfortunately, with this particular virus, previewing the message can be enough unless you have a recently fixed version of Microsoft Outlook / Outlook Express installed. If you are running a Windows machine with windows 98 or 2000 on it, you should have a "windows update" icon on your start menu. If you click this, you go to a microsoft website. Click on "Product updates". This should then give you a list of available updates. You should check this periodically, and install anything that Microsoft labels as a "critical update". This is where they put their security patches that close the loopholes that viruses like this exploit.
Unfortunately, once you have the virus on your machine, doing this isn't enough - you need to get an antivirus program and install it to clean the existing infection - or follow the manual removal instructions that are posted on various websites. This last option is not for the faint-hearted though, as it can get tricky.
Also, although it's not necessary to open an attachment in this case, you should be very wary of opening any attachment that ends in one of .exe, .pif, .scr and so on. Some viruses will send an attachment called something.doc.pif hoping you will see the .doc and assume it's a word document whereas in fact it's an executable program file. I'd recommend not ever directly opening attachments, unless you are absolutely sure that you know what it is and that you were expecting it.
Also, this particular virus does something slightly tricky. If you look at the "from" line in the message, it might say "Joe Bloggs", who you know as "[email protected]" for example. However, if you look at the return email address, it will likely be "[email protected]" - in Outlook click on the from line in the message, right click and choose "properties" to display the underlying email address. So even if you or the anti-virus measure at some company responds saying a message is suspect, or got rejected by a virus filter, the inadvertent sender might not see the response and realise something was wrong.
This seems to have been a fairly major problem in the UK over the last couple of days. The BBC are reporting it now too - here. There is also a fairly technical explanation of it at messagelabs, which is one of the services corporations use to scan their incoming email.