Mathematically, precision and recall are defined as follows:
precision = TP/(TP + FP)
recall = TP/(TP + FN)
recall = TP/(TP + FN)
In our table example, precision and recall are as follows:
precision = 2/2+1 = 0.67
recall = 2/2+2 = 0.5
Out of all the possible options that were indicated as true, precision conveys what fraction of the result was actually correct.
Out of all the possible examples, recall conveys what fraction was picked correctly by the classifier.