Summarize:

Tap the Instagram app icon, which resembles a multicolored camera front, to do so. This will open your Instagram homepage if you're logged in. If you aren't logged in, enter your Instagram account's email address (or username, or phone number) and password before continuing. Scroll down through your homepage feed to find the post on which you want to leave a comment. You can also tap the search bar at the top of the page, enter a user's name, and tap their name to go to their profile if you know who posted the photo or video. It's below the photo or video on which you want to comment. Doing so opens a new page with a comment box. If the person who posted the photo or video has disabled comments, you won't be able to post a comment here. If you want to reply to an existing comment, tap the comment to which you want to reply and then tap Reply instead. You'll do so in the text box at the bottom of the screen. It's on the far-right side of the text box. Doing so will post your comment for the post creator and any of their followers to see.
Open Instagram. Go to a photo or video that you want to comment on. Tap the speech bubble icon. Type in your comment. Tap Post.