yes you will need to download the drivers, there should be a driver disc with the printer but if you cant find it then look here on the HP site
http://h10025.www1.hp.com/ewfrf/wc/softwareCategory?product=412187&lc=en&cc=uk&dlc=en&lang=en&cc=uk
choose the version of Windows you have on a new laptop it is presumably W7, but you need to be sure that you choose either the 32 bit or 64 bit download to match what is installed.
To find out which one you have press the Windows Logo key on the keyboard and the pause/break button and in the section marked System it will specify if it is 32 or 64.
Once downloaded, just install and it should hopefully find the printer and put it in as the default device and all is well.
You should be proud of your Aunt, 80 years young and still willing to learn new things, my mum is the same, (75 years young) and just bought an Acer All in One Quad Core 23" Touchscreen system and loving it, a few lessons from me on the changes from XP to W7 (and from Office 2003 to 2010) and she is surfing away like a teenager.
EDIT >>>>> just read micksmixxx answer and he is quite correct if your Aunts machine is Windows 7 then W7 does have drivers built in for this model, if they don't load when you plug it in then go to the Windows Update (with the printer connected via USB and powered on) and the update should find and install it for you. It will most likely be in the Optional Updates section of the Windows Update site so look for it there.