To expand on this a small bit:
You run calculations on the training set to determine various coefficients.
You can then use the testing set to check how well the predictions do on a wider set of data, and that gives you information about false positives and false negatives.
You can use those accuracy figures to go back and re-train. You do not need to use the same division of training and test data each time: there is a common technique called "leave one out" where you deliberately drop one item at a time from the training set and re-calculate, in case that one was an outlier that was preventing getting a good overall result.
There is a nasty problem in doing classification called 'Overtraining": the calculations might fit the data you have on hand extremely well but be useless for anything else. Dividing into training and testing reduces this risk: if the algorithm has not seen a bunch of data in its calculations then it is not going to adjust itself to be exactly right for that data and bad for other things. Using all of your data to train with is therefor not a good idea.
After the program has gone back and forth on training sets and validation sets, and has decided on the best coefficients, where the data was allowed to affect the algorithm, then it is time to run it on the remaining data and produce a report. The rest of the data might not have a known classification, but it might. If the classifications are known then when the programmer looks at the report the programmer might decide it is time to change the program. Or might not. The report is the kind of thing that gets written up in a paper: we did this and that and with a limited subset of data to train and test with, we did this well on real data. Or perhaps you send it to the people designing the equipment and experiments so they can see what needs to be improved on their end. Eventually you publish the paper or write a report or the like, and other people read it and want to use your program too. But they aren't going to do that if you haven't established evidence that it is not over-training on the particular data you gave it -- and seeing how well it did on data that was not used to design the details of the algorithm is evidence.