From this link:
You can use the 2013 version of Office until September 2016. At that point updates will end (including security patches). So, consider September 2016 your deadline for upgrading.
If you did an "unmanaged" installation of Office 2013 from portal.office.com, it will be automatically upgraded to the 2016 version in February 2016 (unless you set Group Policy to inhibit this, see the comments below). Users can also go to the Office portal and download/install the new version on their own (currently only for people opted into First Release).
If you deployed Office 2013 with the Click-to-Run Deployment Tool, you have more control of when the switchover happens. In this case, administrator involvement is required for the switchover, and it must be completed before support ends September 2016.
If you are using the Deployment Tool:
If you used the Deployment Tool for 2013, there is a new version of the tool for 2016. It is similar to the previous version, and uses the same XML configuration (there are some new options, see below). You must switch to the new version to deploy 2016.
Note the new version version of the tool is for 2016 only. If you plan to support both versions (e.g. keep using 2013 while you do test deployments of 2016), you will need to keep the deployment tools and installation files for both versions side by side on your Application Deployment file share.
The new deployment tool also has support for a "fast" track for updates (First Release) and a "slow" track for updates (Current Branch for Business). You specify this in the XML configuration and you can modify it after installation via Group Policy (or the underlying Registry setting).
All of the above info only applies to Office 365 customers that are licensed for ProPlus (e.g. the E3 plan), and installed via Click-to-Run. If your Office desktop install is not tied to an Office 365 subscription (e.g. Volume License or Retail), things are different.
Something to keep in mind:
If you use Group Policy or logon/startup scripts to apply standard Office settings for your users, note that you will need to have Office 2016 versions of your policies prepared for deployment. Microsoft just released the ADMX templates for Office 2016. They also released a script to copy your settings from a previous version.