Here's how drops will look like in the following views:-
The number of unread drops in your river is normally indicated on the right hand corner. Clicking and viewing a drop will automatically mark it as read.
To mark a drop as read (without having to view it), hover your mouse over the drop, and click on the tick icon.
Your "unread" drop count should decrease, and your "read" drop count should increase on the right hand menu bar.
{More to be added}
To delete a drop from your river, hover over a drop, and click on the "X" sign