To use Periscope, you must first download the app. Initially, Periscope required a Twitter account for signup/login, and you can still log in that way, but now if you don't have a Twitter account (or just don't want to use it for whatever reason), you can also sign up using your phone number.
When you first open the app, it goes to the main watch screen, also accessible via a TV icon link at the bottom of the screen. This page lists livestreams of the people you are following (if any) and their recent broadcasts from the last 24 hours, including anything you've streamed recently. You simply touch any of the livestreams to start viewing immediately, and touch the X in the upper right to end viewing.
Aside from the TV icon, other bottom-screen buttons include a globe that takes you to the Global page, a camera that takes you to the Broadcast Now page and a cluster of people that takes you to the People page.
The Global page lists live streams from users all over the world. As with the main page, just touch one of the items on the list to begin viewing the stream, and hit the X at the top right to exit back to the previous screen.
While viewing any live broadcast from either the main or Global page, a stream of chats will scroll up from the bottom left, and you can participate if you like. Just type some text in the window that says "Say something..." and hit "Send." If a lot of people are chatting, you might see a lock icon and the text "Broadcast too full" in the chat window. If you click it, you'll receive a notification that there are a lot of people watching and only early joiners can chat. The app can also reportedly revert to only allowing users the broadcaster follows to chat once a large enough number of people are viewing.
You will see animated, color-coded hearts that you or others are sending rising from the bottom right. You send them by tapping the screen. Hearts are the equivalent of the Facebook like, except that you can send as many as you want. In Periscope, your popularity is ranked by number of hearts received, not number of followers.
To bring up more information while viewing a stream, swipe to the right or click on the person icon on the bottom right that shows number of people viewing at the moment. Either action brings up an information screen with the title of the stream, the broadcaster's location on a map (if allowed), the streamer's name and handle, a follow icon, and "Share" and "Hide Chat" (or "Show Chat," if you've already hidden it) buttons. Below that, all the current viewers are listed, as well as the number of viewers. At the bottom are a "Report Broadcast" icon and link. "Hide Chat" is useful if you find that the chats are obscuring your view of the video. The "Share" button allows you to suggest the current stream to your followers (either selected ones or all of them) via a push notification. If you get one of these notifications while you're in the app, you can either dismiss or view the suggested stream with a touch. And "Report Broadcast" is for reporting inappropriate content.
If you select a stream that's already ended, you can hit the play icon to start viewing the older stream. You'll see the video, the chat stream and the hearts, although you won't be able to chat. You can, however, send hearts while watching a replay. These videos will disappear 24 hours after broadcast.
The Broadcast page allows you to start streaming whatever your phone camera is seeing (and whatever your microphone is hearing). We'll elaborate on broadcasting in a bit.
The People page lets you select who to follow. The first users on the list are people you follow on Twitter who are also on Periscope, and the people below that are the "Most Loved" Periscope users. At the top left, there is a search icon to help you find people by name or username, and at the top right, there is a single person icon that takes you to your own user profile page.
Your profile page shows your username, the number of hearts you have accumulated, your bio (from Twitter, if you don't edit it within the app) and links to lists of who you are following, who is following you, users you have blocked and your past broadcasts. You can also adjust a couple of settings there and get to the Help Center, Terms of Service and other useful information. A logout link is at the bottom of this screen.