Data science is a discipline that works with machine learning and Big Data, as well as many other things. I work as a Data Scientist, and while I do use machine learning and Big Data in my job, it is not all I do. Also, you need to consider that there are different types of data scientists.
Machine Learning is, at its most basic, a predictive model created by feeding it data. Let us imagine we have a list of houses that sold recently. We have two columns, one with the square footage of the house and one with the price the house sold for. We could feed this data into a machine learning algorithm and it will build a model for us. Now if I ask the model how much a 2000 sq ft house will sell for, the model will provide us price based on the list of prices we had given it. Now obviously we use much more complex data sets with many many more variables, but at the end of the day machine learning boils down to asking a computer to either classify an object (is the picture a cat or dog?), provide a numeric value (regression - think of the house price example), or cluster (see how data should be best grouped based on attributes - think of all the students in your high school and how they can be grouped: jocks, drama clubs kids, nerds, popular crowd).
Big Data is massive, fast moving, data sets. It is a popular term, but not all data science or machine learning involves Big Data. Twitter is great example of big data with millions of tweets every few minutes.
In my case, I am what you might call an operational data scientist. I work in financial compliance at Verizon helping to hunt down people who are "gaming the system" or stealing from us by using loop holes in our policies. The biggest part of my job is finding, gathering, and cleaning data so I can analyze it. Once I have the data I may run it through a machine learning algorithm to create a predictive model they may help us to predict which people we should look at more closely (make the haystack a little smaller - easier to find the needle in.
A big data example I worked on with another company was using the voice recordings or people calling customer service. I was able to determine certain speech patterns that were more likely to be used by someone trying to commit some type of fraud. We were able to use this information to alert the customer care reps who to be on the look for.