What does the expression push and pull mean in the following article from The New York Times?
This is not a phrase which bears a clear meaning on its face. 'Push' and 'pull' are often employed to express repulsion and attraction, and 'gender' seems to be exerting such repulsion and attraction; but beyond that you have to read the article to find out exactly what is meant. That in fact is the purpose of a headline: to intrigue you, to make you want to read more.
Eventually you will find the phrase fairly explicitly explained:
Each day in Mecca provided powerful reminders of a religion that seems to simultaneously embrace women and push them away.
There's a lesson in this for learners (and indeed for students of any topic): don't be in too big a hurry to achieve complete understanding on your first reading. Often you learn more by running quickly through an article or story, to get a rough sense of the overall content, and then re-reading to fill in the fine details. If you practise reading inferentially—if you fill in the gaps of your understanding as much as possible by figuring out what a given word or expression has to mean for the passage to make sense, without immediately rushing to a dictionary or to ELU—you will eventually find reading English much easier and more natural, and a lot more pleasurable.