The Publishing Queue is where you can see the status of items you've published or scheduled to publish, and also where you can verify if someone else has already published (or unpublished) something.
To open the Publishing Queue, look for the Publishing Queue icon in the top ribbon.
Click on that icon, and the Publishing Queue window will open.
Initially, the window will be blank. Click the Show Tasks button, and this will display the items you've either published today or scheduled to publish.
The queue does not update in real-time, so if you are monitoring an item as it moves through the publishing queue, you will need to either keep clicking "Show Tasks" or click the refresh icon located next to the words "Publishing Tasks" on the middle left of the window.
Tip: If your items appear stuck in the Publishing Queue, for the User field at the top, select "All" from the drop-down and click Show Tasks again. This will show you what everyone using Tridion has published today. You can see what is ahead of your items in the queue, and watch the queue as it processes through all the items (you'll need to keep clicking Show Tasks to watch them change status). This is one way you can tell if the queue is truly stuck, or if it's just churning through a lot of items at the moment.