Big data refers to extremely large data sets that may be analyzed computationally to reveal patterns, trends, and associations.
Data mining refers to the process of discovering useful patterns in very large databases. It uses methods from statistics, machine learning, and database management to restructure and analyze data to extract knowledge or information from the data. (Encyclopedia of measurement and statistics)
A data scientist is somebody who works with a data set and is using it to discover an answer to a question or something new. This is a quickly growing and expanding field, and includes people with specialties in data collection, management, visualization, or with working with special types of data such as big data.
Different methods of working with data include data mining, working with "big data" etc.
Data scientists will also usually learn a programming language, such as SQL, or be familiarized with software specific to working with data.
If you are interested in learning more about what it takes to become a data scientist, there are many free courses available on Coursera!
Contemporary Issues in Exploratory Data Mining in the Behavioral Sciences by 
