For a quick introduction, you can read this AMS survey.
* _What is … Persistent Homology?_ by Shmuel Weinberger
A basic notion in persistent homology is a _barcode_. The following article gives an introduction to the subject with an emphasis on shape recognition, and tells you what a barcode its.
* _Barcodes: The Persistent Homology of Data_ by Robert Ghrist
Here is another introductory survey article giving you more background material about the theory and implementation of persistent homology. This one also talks about some heavier stuff from algebraic topology, like spectral sequences.
* _Persistent Homology – a Survey_ by Herbert Edelsbrunner and John Harer