Both Carlsson's and Ghrist's survey papers offer a very good introduction to the subject

• Topology and Data by Gunnar Carlsson
• Barcodes: The Persistent Topology of Data

• Extracting insights from the shape of complex data using topology A good introductory paper in Nature on the Mapper algorithm.
• Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition A more technical presentation of Mapper.
• Topological Data Analysis and Machine Learning Theory Applications of TDA to machine learning.
• Ayasdi, the company founded by Gurjeet Singh and Gunnar Carlsson, have several good videos and whitepapers on how they use Mapper and TDA in machine learning pipelines.

• Computational Topology: An Introduction A good introducgtory book on persistent homology
• Elementary Applied Topology A book by Robert Ghrist that goes beyond applications of algebraic toplogy to data analysis, but a very good read. There is a very inexpensive print version and the PDF is available for free.