Learning about them: https://en.wikipedia.org/wiki/Support-vector_machine.

That will tell you SVMs are ancient (linear version dates back to 1963), and that what they do here isn’t really machine learning, but something similar to linear regression: just as linear regression finds the best (in some strict mathematical sense) line describing a set of points, this finds the best (in a similar mathematical sense) line splitting two sets of points.

For software, take a look at https://www.csie.ntu.edu.tw/~cjlin/libsvm/. Easy to use, fairly flexible, with a Java applet you can play with.

SVMs are as old school ML as they get. They guarentee the maximum separation at the decision boundary. However it doesn’t scale very well for higher dimensional data. The standard used to be to use some dimensionality reduction technique like PCA to preprocess before feeding it into the SVM.

This is all before deep learning.

Exactly. Perhaps the paper could have given a clearer message if the abstract had characterized SVMs as a quadratic optimization technique instead of as machine learning?