Hate speech is plaguing the cyberspace along with user-generated content. Adding counter speech has become an effective way to combat hate speech online. Existing datasets and models target either (a) hate speech or (b) hate and counter speech but disregard the context. This paper investigates the role of context in the annotation and detection of online hate and counter speech, where context is defined as the preceding comment in a conversation thread. We created a context-aware dataset for a 3-way classification task on Reddit comments: hate speech, counter speech, or neutral. Our analyses indicate that context is critical to identify hate and counter speech: human judgments change for most comments depending on whether we show annotators the context. A linguistic analysis draws insights into the language people use to express hate and counter speech. Experimental results show that neural networks obtain significantly better results if context is taken into account. We also present qualitative error analyses shedding light into (a) when and why context is beneficial and (b) the remaining errors made by our best model when context is taken into account.