Do I have to purchase Word again after getting a new computer?

It depends entirely on how you bought Office 2007 on the old machine. If you bought Office in the big ugly yellow plastic CD box, then you almost certainly have a license that allows you to install on 2 or 3 machines and to transfer those installations to new computers.

If you bought it factory installed AND activated, then you have an OEM license and you do not have the right to transfer it to a new machine.

If you bought it as an online download probably have a license that does not allow you to transfer to new devices.

Simplest thing is find the product key. Then install and internet activate on the new machine to see what happens.  If the internet activation fails you can try a phone activation. When you phone in, don't reply to the initial prompts. Just wait, a long time, until you are offered the option of talking to a live "service" rep.  Explain the situation to them, they can tell you what type of license you have to explain why the activation failed.

Regardless, Office 2007 will be going off support by MS in 1 year time. At that time you really should update to a newer version, so it may be just as simple to update now.

If you do decide to update now you have a few decisions to make. Do you want/need a 365 pay-forever "subscription" or a 2016 one-time payment License.  Then you have to decide which specific bundle you want in the specific license category you decide on.

http://www.microsoftstore.com/store/msusa/en_US/cat/categoryID.68155000

HomeBusinessStudent

Microsoft Office and Office 365 give you access to the programs you need to work, play, and stay connected. From the power to create documents and presentations, to the ability to design database apps and stay in touch via email, Office simplifies the way you work online, and off.

What you might not realize, is that Office comes in a variety of versions designed for different needs, uses and computers. But before you buy, let's compare Microsoft Office versions so you can choose the right software for all the things you do.

Office 365 Home: Designed for use on up to 5 PCs or Macs, as well as 5 tablets – including iPad, Android, or Windows, and 5 phones. In the package, you'll receive access to Word, Excel, OneNote, PowerPoint, Outlook, Publisher and Access. In addition, you'll automatically receive the latest Office updates (including Office 2016), and get 1TB of OneDrive cloud storage, free Microsoft Answer Desk support and 60 minutes of Skype calling per month.

Office 365 Personal: When you compare Office versions, Office 365 Personal gives you all the same great features you'll find in Office 365 Home, but on just one PC or Mac and one tablet device, instead of up to five of each.

Office 365 University: Of the versions recommended for students, Office 365 University offers all the same great features in Office 365 Personal, but for 2 PCs, Macs, or tablets and two phones.

Office 2016 Home & Student: Microsoft Office 2016 features new, modern versions of Word, Excel, PowerPoint, and OneNote that are designed to help maximize your productivity and give you rich authoring features, design controls for pixel-perfect layouts and intuitive tools to help you make the most of your data. Plus, because you'll have access to your docs in the cloud whenever you need them, you can work when, how and where it suits you best. And for those times when collaboration is critical, you can even share documents and get your team on the same page. (Designed for use on one PC.)

Office 2016 Home & Business: With the same great features as Office 2016 Home & Student, as well as Outlook, Office 2016 Home & Business gives you everything you need to get more done -- and stay connected with friends, family and colleagues around the globe.

Office 2016 Professional: Now you can do your best work – anywhere, anytime and with anyone. In addition to the 2016 versions of Word, Excel, PowerPoint, Outlook, and OneNote, Office 2016 Professional includes new iterations of Publisher and Access, which can help increase your productivity so you can do more in less time. Best of all, with your documents stored online, collaborating with teams around the world is easy.

In general, when you compare Office vs. Office 365, the most important thing to note is that Office 365 is always up to date, so you'll never have to purchase upgrades or wonder if you're using the most recent version.

[1] Office 365 is compatible with Windows 7 or later, and Mac OS X 10.10. Application availability and features that come with Office 365 vary by platform and device; current Office application versions for Office 365 are Office 2016 for Windows and Office 2016 for Mac. Visit www.office.com/information for full system requirements, including compatible tablets and smartphones for Office 365.

[2] Available on PC only.

[3] Skype available only in select countries; see FAQ for details. Calling to select countries only. Excludes calling to special, premium and non-geographic numbers.

https://support.office.com/en-us/article/What-s-the-difference-between-Office-365-and-Office-2016-ed447ebf-6060-46f9-9e90-a239bd27eb96

http://www.techrepublic.com/article/how-to-install-office-365-in-os-x-el-capitan/