"Training data" and "testing data" refer to subsets of the data you wish to analyze. If a supervised machine learning algorithm is being used to do something to your data (ex. to classify data points into clusters), the algorithm needs to be "trained".
Some examples of supervised machine learning algorithms are Support Vector Machines (SVM) and Linear Regression. They can be used to classify or cluster data that has many dimensions, allowing us to clump data points that are similar together.
These algorithms need to be trained with a subset of the data (the "training set") being analyzed before they are used on the "test set". Essentially, the training provides an algorithm an opportunity to infer a general solution for some new data it gets presented, much in the same way we as humans train so we can handle new situations in the future.
Hope this helps!