An svm is a type of linear classifier. If you don’t mess with kernels, it’s arguably the most simple type of linear classifier.
Linear classifiers (all?) learn linear fictions from your data that map your input to scores like so: scores = Wx + b. Where W is a matrix of learned weights, b is a learned bias vector that shifts your scores, and x is your input data. This type of function may look familiar to you if you remember y = mx + b from high school.
A typical svm loss function ( the function that tells you how good your calculated scores are in relation to the correct labels ) would be hinge loss. It takes the form: Loss = sum over all scores except the correct score of max(0, scores – scores(correct class) + 1).