Data mining deals with searching large amounts of data for very specific information. You are searching through your data looking for something specific. For example, a credit card company would use data mining to learn about buyers habits by analyzing purchases and their locations. This information then becomes very useful for things such as targeted advertisements.
Machine learning, on the other hand, focuses on performing the actual task of searching for that data using algorithms you have provided. Makes sense?
Enough said for now, but here is an excellent link where you can learn more about data mining: https://blog.udacity.com/2014/12/24-data-science-resources-keep-finger-pulse.html