I have a ZTE N817 and I bought a 16GB sd card for it and when I try to download something, all it says is that there is "insufficient storage on device". What should I do to get my phone to register that I have am sd card full of untouched storage?
That is because your device downloads data in your device internal memory by default and you don't have sufficient space on your device internal memory . In this case you can transfer some data to your sd card such as media files or change download location to sd card.
When you download apps and updates to apps to your device, your internal memory is used as temporary storage for the downloaded APK which is then installed once it has finished downloading; you will need to clear some internal storage first. Caron was spot on about that.
However, if you are running Android 5 or higher, there is no way to treat the external SD card as storage for installed apps (until Android 6, which allows the external SD card to be treated as extended internal storage via Adopted Storage); prior versions of Android (at least Kitkat, someone please correct me) did allow apps to be moved to the external SD card.
Looking at your phone model, it does look like your phone is running KitKat, so you're in luck! All you need to do is free up space on your internal storage. What I would do first is go into the Settings app, then go to your installed apps list, then sort by size. From there, tap on an app then tap on "Move to SD card". If the app is a user app and the developer has allowed it to be installed on the SD card, the button should be available to tap, otherwise it will be greyed out.
Apps on the SD card won't suffer from too much of a performance hit but if the SD card is removed you won't be able to use the apps installed on it.