Apps like to let you know when something has cropped up which requires your attention.
They do this by displaying a notification to you. There are two primary ways that notification get shown to you, these are banners and badges.
Banners are short full width containers that appear on your screen, each one with information which tells you what created the notification, and a brief description of what the notification is about.
When a banner first appears, depending on whether your device is locked or not, the banner will either appear on your lock screen or it will pop down briefly from the top of your phone before hiding away.
A badge is a small circle which appears on top of an apps icon.
The badge will contain a number which correlates to the number of notifications that the app has for you.
To clear the notification go onto the app and view what was causing the notification, which changes from app to app.