Gives you a central hub to handle support tickets (in my case, internal). Rather than handling issues faced by colleagues through e-mail or a chat program, thus, cluttering either up, you can sort it all away into a separate place. It usefully has both e-mail notification and Slack integration abilities, which means being aware right away when an issue is sent in. It also has the ability to link tickets from JIRA, so if an issue requires additional intervention from other teams, you now have that connection. Also useful is if you have some desire for anonymity (obscure/black box who specifically is handling issues), this will work far better than e-mail or Slack in providing that. There are also additional features such as stats and metrics, along with the ability to add tags to tickets, required fields, and macros to automate certain workflows. Again, if you are working extensively with handling issues of all manners, you will find Zendesk quite useful for you.
While the software does a good job of providing a central hub for managing support tickets, it has a couple of improvement points. The ability to not reopen a ticket if someone is simply thanking you would be nice, and one that has been requested a while that has never been implemented. Navigating the help center where you can write articles is a bit confusing to navigate and requires specific flows to get to view vs edit mode. Furthermore, the formatting tools you are offered for creating articles are quite limited, clunky, and awkward. If you want to write anything other than very bland and simple text, you'd be better off linking to an external site with better options, such as Google Docs.