Open @BotFather in Telegram and send /newbot command. Pick your bot name and username:

@BotFather will send you a unique string called bot token:

Keep this token in secret! 

Your bot is now created. To set bot's profile picture, send /setuserpic command to @BotFather followed by an image:

To change the "about" text shown in bot's profile, send /setabouttext command:

Now, the bot's profile has an avatar and a short description in the "about" section:

You may also want to change the bot description (shown in the bot dialog before clicking START). To do that, use /setdescription command in @BotFather.