Many social network applications depend on robust representations of spatio-temporal data. In this work, we present an embedding model based on feed-forward neural networks which transforms social media check-ins into dense feature vectors encoding geographic, temporal, and functional aspects for modelling places, neighborhoods, and users. We employ the embedding model in a variety of applications including location recommendation, urban functional zone study, and crime prediction. For location recommendation, we propose a Spatio-Temporal Embedding Similarity algorithm (STES) based on the embedding model. In a range of experiments on real life data collected from Foursquare, we demonstrate our model's effectiveness at characterizing places and people and its applicability in aforementioned problem domains. Finally, we select eight major cities around the globe and verify the robustness and generality of our model by porting pre-trained models from one city to another, thereby alleviating the need for costly local training.