Refer to this sample Precision-Recall curve plotted for evaluating the classification performance of 3 different algorithms on the same test set. Precision is plotted on the Y-axis and Recall on the X-axis, percentages have been expressed in decimals.
The curve is plotted by steadily increasing the cutoff value from 0 on the right-most side (recall=100%) to the highest value on the left-most side of the graph (recall=0%). For simplicity, we'll refer to the algorithms as "red", "green" and "blue".
Now, when we compare points A and B, the recall reduced by 20% when the cutoff was increased from point A to point B, while the precision increased by 4% since false-positive rate was reduced. But the corresponding precision for algorithm "green" was much worse indicating a poorer classification performance. When we examine the performance at point C, the "red" algorithm performs much better (higher precision) than the other two for the same level of recall.
Based on the nature of the classification task and the "cost" of false positives vs. false negatives, an appropriate algorithm may be chosen.
For example, if a recall of 30% is the minimum acceptable for the task at hand, then algorithm "red" should be chosen since it has the highest precision, i.e. it will pick up more true positives for the same number of predicted positives.
Lets ignore algorithm "red" for a moment and consider only the "blue" and "green" classifiers. If the minimum acceptable recall is 70%, then algorithm "blue" would be chosen since it is more precise at that rate of recall. But if the minimum acceptable recall is 20%, then algorithm "green" should be chosen since it has higher precision at that recall rate.
It is possible for an algorithm to have better precision and recall than another algorithm and this can be indicated by plotting their curves together. For the same algorithm, usually if cutoff is raised precision increases while recall decreases. However, a lot depends on the algorithm being used and how well is it able to classify the dataset based on the attributes provided.