Score in machine learning is a metric to measure how good is a model. For classification problems, we use accuracy, AUC, F1 score, etc. For regression problems, we use RMSE, AIC, BIC, etc.
F1-score combines the precision and recall of a classifier into a single metric by taking their harmonic mean.
A more general F score called F-beta uses a positive real number beta, where beta is chosen such that recall is considered beta times as important as precision.