If you already have Office installed on your 'old' PC and you have the CD then you should be able to install it on your new one. I believe that the license agreement allows installation on up to three PCs.
Student and teacher is the full version sold at a discount to people whio can claim to be one of these. When you buy this version , initially all you get is a box conatining a certificate which you have to have signed/stamped by an educational establishment. You send that off and receive in the post the full set of disks that you otherwise would buy in the shop at considerably higher cost. So....if you are attending an evening class; have a child at school/university/etc or otherwise have a connection to an educational establishment so that you can get the signature/rubber stamp, you can buy the "student and teacher" version.