Push notifications are notifications directly sent from a server to your phone using Google Cloud Messaging or similar, without an app on the phone having to run all the time to create the notification
Many apps need to run all the time or periodically to get new data from a server in order to notify you of it: for example, an email client might run every 15 minutes to check for new mail on the server. This is called client pull or polling.
This is bad for battery life, but there's another way: when there's a server, it can send a message directly to the phone to give an app new data or trigger some action (such as updating an app). This is called server push, so when this is used to generate notifications on the device, they're called push notifications. (See How do push notifications work? for more details on how this process works.)
Note that although SMS messages and phone calls are sent without the phone having to poll, these aren't usually referred to as push notifications, because they're special and don't use Internet.
If your question is about notifications in general, or specifically about apps which notify you by polling for new data, use the notifications tag instead. Use this tag only if your question is specific to notifications pushed from a server using Google Cloud Messaging or similar.