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If you have been searching the Instagram app on your iPhone for a delete button, you will not find one - Apple's App Store guidelines require account deletion to be reachable, but Instagram only implements it through its website, not the app itself.
On your iPhone, open Safari (or any other browser) and go to instagram.com. Log in with your username and password if you are not already signed in.
The mobile version of instagram.com in Safari sometimes hides this option deeper in the menu than the desktop layout does. If you cannot find it, try tapping Request desktop website in Safari's menu (the "aA" icon in the address bar) to load the full desktop layout, where the setting is easier to locate.
Deleting your account and deleting the app are two separate things. Removing the Instagram app from your iPhone (press and hold the icon, then Remove App) does not delete your account - it just removes the app itself. You still need to complete the steps above first.
If you just want a break, deactivating instead keeps everything intact and reversible. See our guide on how to deactivate Instagram temporarily, or read the full deactivate vs. delete comparison first.