Before you can use the new feature, you'll need to upgrade to version 3.5 of the app on your iPhone or Android device. Once you are running the latest version of Instagram, you will see an Add People line on Instagram's Share screen after you have cropped and filtered your photo.
Tapping on the Add People line opens a screen asking you to tap the photo to add people. When you do this, a small tag appears, asking Who's this? You can drag this tag around to line it up with the person or people in the photo if your initial tap was inaccurate. A search bar at the top of the screen lets you search Instagram for the right user. It autocompletes with your followers but lets you tag anyone (or any brand) on Instagram.
From your feed or profile, a single tap on a photo will hide and unhide any photo tags. If you'd like to remove a tag that someone added of you in a photo, tap the tag and then tap More Options. Here, you will find options to remove yourself from the photo or report the photo as inappropriate.
I am certainly glad that I went to college during a time that predates social networks for reasons like this: the Photos of You feature is set up so that photos in which you are tagged will automatically appear in your profile. If you are wary of your profile filling up with photos you'd rather not share with the rest of your Instagram community, there is a way to set it so photos in which you are tagged show up only after your approval.
To do so, go to your profile and tap the Photos of Me button (it's the right-most button in the belt below the Edit Your Profile button). On the Photos of You screen, tap the Settings button and choose Add Manually.
Instagram must know that many of its users are wary of this new photo-tagging feature; Photos of You won't be made public until May 16, giving you two weeks to try out the feature and adjust your privacy settings accordingly.