Area under the ROC curve is one of the most useful metrics to evaluate a supervised classification model. This metric is commonly referred to as ROC-AUC. Here, the ROC stands for Receiver Operating Characteristic and AUC stands for Area Under the Curve. In my opinion, AUROCC is a more accurate abbreviation but perhaps doesn't sound as nice. In the right context, AUC can also imply ROC-AUC even though it can refer to area under any curve.

With a ROC curve, you're trying to find a good model that optimizes the trade off between the False Positive Rate (FPR) and True Positive Rate (TPR). What counts here is how much area is under the curve (Area under the Curve AuC). The ideal curve in the left image fills in 100%, which means that you're going to be able to distinguish between negative results and positive results 100% of the time (which is almost impossible in real life). The further you go to the right, the worse the detection. The ROC curve to the far right does a worse job than chance, mixing up the negatives and positives (which means you likely have an error in your setup).