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Its a purple-red-orange gradient app with a white, square, camera icon. It's below the blue Login button. You may have to tap Sign in at the bottom of the screen first. It's on the middle-right side of the screen. On Android, tap Send an SMS. Doing so will send a text with a link in it to your provided phone number. You'll tap → in the top-right corner of the screen instead if you're using an Android device. You should have a text from a five-digit number (formatted like 123-45). The text's body will say "Tap to get back into your Instagram account:" followed by a link. You may have to tap it twice You'll see this option at the bottom of a "Open this page in Instagram?" dialogue window. Doing so will open your Instagram account.  If you're using an Android device, tapping the link will take you to a password reset page instead. You'll be able to type in a new password twice and tap Reset password before being redirected to your Instagram account in a mobile browser. At that point, you can open Instagram and log in with your new password. If you'd like to reset your iPhone's Instagram password but don't have access to your old email account, you'll need to tap Edit Profile on your profile page and enter a new email address into the email field. From there, you can tap Done in the top-right corner of the screen, tap the settings gear, and tap Reset Password. You'll receive a link in your new email address.
Open Instagram. Tap Get help signing in. Tap the Phone tab. Tap Send Login Link. Open your phone's messages app. Tap the message from Instagram. Tap the Instagram link. Tap Open.