Not a lot, if it's just to get started. I started off with the John Hopkins Data Science Specialisation by Coursera, which requires very little math. The course website says: "We also suggest a working knowledge of mathematics up to algebra (neither calculus or linear algebra are required)." The next one, Machine Learning by Andrew Ng / Stanford required some basics of linear algebra, but provided an optional module for those who did not have such a background. This covered matrices and vectors, matrix multiplication, inverse.
So the math isn't really difficult if you just want to cover the basics. I should point out one caveat, however. The Coursera Machine Learning course is really a watered down version of the original CS229 from Stanford, where the math is considerably more involved (just take a look at the review notes on linear algebra, probability, convex optimization, etc. to see what they go into). I took a somewhat similar course, Learning from Data by Caltech/edX (see the math from the slides / notes here) and the math got pretty tough, especially at the later part of the course.
Of course, its possible (and likely) that many parts of data science / machine learning do not require such advanced techniques. In many cases, it may be possible to get away with a "black box" approach - i.e. treat the tool as a black box, and not worry too much about what goes on inside the box. However, out of personal interest, it may be worthwhile to understand a bit more about how such algorithms work. Furthermore, who knows whether at some point in the future, you may run into issues with debugging algorithms that require some deeper sort of fundamental knowledge?