Summarize this article:

Tap the YouTube app icon, which resembles a red YouTube logo on a white background. This will open your YouTube main page if you're logged into your YouTube account. If you aren't logged into your YouTube account, you'll need to select your Google account (or add a new one) and then sign in with your email address and password before proceeding. It's in the top-right corner of the screen. Type in the name of a video on which you want to leave a comment, then tap Search (iPhone) or Return (Android). Tap the video on which you want to comment. This will open the video. This section is below the list of related videos. You'll see it near the top of the "Comments" section, just right of your Google Account's profile picture. Your phone's keyboard should appear on-screen. If you want to reply to an existing comment instead, tap the comment. Enter the comment that you want to leave on the video. It's a blue, paper plane-shaped icon in the lower-right side of the comment field. Doing so will post your comment on the video.

Summary:
Open YouTube. Tap the "Search"  icon. Search for a video. Select a video. Scroll down to the "Comments" section. Tap the "Add a public comment…" text field. Type in your comment. Tap the "Send"  icon.