Organizations are increasingly interested in allowing external data scientists to explore their sensitive datasets. Due to the popularity of differential privacy, data owners want the data exploration to ensure provable privacy guarantees. However, current systems for answering queries with differential privacy place an inordinate burden on the data analysts to understand differential privacy, manage their privacy budget, and even implement new algorithms for noisy query answering. Moreover, current systems do not provide any guarantees to the data analyst on the quality they care about, namely accuracy of query answers. We present APEx, a novel system that allows data analysts to pose adaptively chosen sequences of queries along with required accuracy bounds. By translating queries and accuracy bounds into differentially private algorithms with the least privacy loss, APEx returns query answers to the data analyst that meet the accuracy bounds, and proves to the data owner that the entire data exploration process is differentially private. Our comprehensive experimental study on real datasets demonstrates that APEx can answer a variety of queries accurately with moderate to small privacy loss, and can support data exploration for entity resolution with high accuracy under reasonable privacy settings.