> I have read that the Buddha said the cause of suffering is ignorance
> of the "nature of reality". Is this correct?

Coming into existence or 'birthing' (jati) is the cause of suffering (dukkha). 
[Dependent Arising - Paticca samuppāda][1] by [Piya Tan][2]

> What is the "nature of reality" that the Buddha talks about?

All conditioned things (forms, verbalization, mental thoughts), are forever changing (anicca), lead to suffering (dukka), and devoid of a permanent entity/owner (anattma). Read about the Tilakkhana or the Three Characteristics in the Dhamma. 

may you be happy. 


  [1]: http://www.themindingcentre.org/dharmafarer/wp-content/uploads/2009/12/5.16-Dependent-arising-piya.pdf
  [2]: https://en.wikipedia.org/wiki/Piya_Tan