Link Prediction is the problem of predicting which links will appear in a network. For example, we can have a friendship graph in Facebook or another social network, and functionality like people you may know is an application of Link Prediction. So, we can see Link Prediction is a recommendation system for social networks.
For this problem, we need to find a dataset that contains a graph evolving over time. Then, we can consider such a graph at some point in its evolution, calculate some characteristics between the existing links, and, based on that, predict which links are likely to appear next. Since for such graphs we know the future, we can use this knowledge for evaluating the performance of our models.
There are a number of interesting datasets available, but unfortunately, most of them do not have a time associated to the edges, so it is not possible to see how these graphs developed over time. This makes it harder to test the methods, but, of course, it is possible...