For some people, the Apple Wallet app in iOS might be synonymous with Apple Pay -- something you only see briefly after double-tapping the home button. For the experienced though, it's also a way of keeping passes handy without stuffing paper copies in your pocket.
There are a variety of ways to add passes, including Wallet-ready apps, scanning a code, or even tapping an iOS notification after using Apple Pay at a store. Often, however, you'll be tapping on an "Add to Apple Wallet" icon, whether on a webpage or in an email, such as a receipt for concert tickets. Be sure to do this from your iPhone to make it work.
If you do need to scan a physical pass, this can be done by tapping the "+" button in the Passes section of Wallet.
After confirming that you want to add an item, it should show up in Wallet automatically. Tapping on a pass there will bring up details and make it ready for use, including any QR or barcodes that need to be scanned at a merchant, station, or venue. In some cases, simply appearing at the right location will trigger a lock screen notification you can swipe to jump into the appropriate pass.
You can swipe down on a pass to go back to the main screen, or swipe left and right if there are multiple items associated with the same event, like a ticket for your spouse.
Each pass may also include a link to a relevant app in the lower-left, but will always include an "Information" button in the lower-right. Here you can do things like toggle lock screen notifications and automatic content updates.
More importantly though, this is the only way to remove a pass, as even time-sensitive ones will clutter up Wallet long after their expiry date. It's worth coming back to the app periodically to clean things up if you become a regular user.