What type of licensing does Esri offer?

Esri offers perpetual and term licensing. 

Perpetual licensing carries optional yearly maintenance that provides you with software updates, technical support, and ArcGIS Online access. If maintenance is not renewed, the software will remain licensed at the last version under maintenance. 

Term licensing is a subscription-based plan that will come with all the benefits of maintenance as long as you renew the subscription. The license will time-out if not renewed.

Read more about license enablement options for ArcGIS Desktop, including Named User, Single Use, and Concurrent Use license enablement options.