What is that red badge with an exclamation mark?
You can find that icon in two different places.
On the log button in the top left corner of the screen
This indicates that there is something new in the Battle Log. Open it and you will find all new items (e.g. a new invasion) also marked with the same indicator.
On the speech bubble button in the top right corner of the screen
This indicates that there is a new unread message in your Alliance Chat. Tap on it to read it.