Error correction techniques have been used to refine the output sentences from automatic speech recognition (ASR) models and achieve a lower word error rate (WER) than original ASR outputs. Previous works usually use a sequence-to-sequence model to correct an ASR output sentence autoregressively, which causes large latency and cannot be deployed in online ASR services. A straightforward solution to reduce latency, inspired by non-autoregressive (NAR) neural machine translation, is to use an NAR sequence generation model for ASR error correction, which, however, comes at the cost of significantly increased ASR error rate. In this paper, observing distinctive error patterns and correction operations (i.e., insertion, deletion, and substitution) in ASR, we propose FastCorrect, a novel NAR error correction model based on edit alignment. In training, FastCorrect aligns each source token from an ASR output sentence to the target tokens from the corresponding ground-truth sentence based on the edit distance between the source and target sentences, and extracts the number of target tokens corresponding to each source token during edition/correction, which is then used to train a length predictor and to adjust the source tokens to match the length of the target sentence for parallel generation. In inference, the token number predicted by the length predictor is used to adjust the source tokens for target sequence generation. Experiments on the public AISHELL-1 dataset and an internal industrial-scale ASR dataset show the effectiveness of FastCorrect for ASR error correction: 1) it speeds up the inference by 6-9 times and maintains the accuracy (8-14% WER reduction) compared with the autoregressive correction model; and 2) it outperforms the popular NAR models adopted in neural machine translation and text edition by a large margin.