One of the key challenges facing genomics today is how to efficiently analyze the massive amounts of data produced by next-generation sequencing platforms. With general-purpose computing systems struggling to address this challenge, specialized processors such as the Field-Programmable Gate Array (FPGA) are receiving growing interest. The means by which to leverage this technology for accelerating genomic data analysis is however largely unexplored. In this paper, we present a runtime reconfigurable architecture for accelerating short read alignment using FPGAS. This architecture exploits the reconfigurability of FPGAS to allow the development of fast yet flexible alignment designs. We apply this architecture to develop an alignment design which supports exact and approximate alignment with up to two mismatches. Our design is based on the FM-index, with optimizations to improve the alignment performance. In particular, the $n$ -step FM-index, index oversampling, a seed-and-compare stage, and bi-directional backtracking are included. Our design is implemented and evaluated on a 1U Maxeler MPC-X2000 dataflow node with eight Altera Stratix-V FPGAS. Measurements show that our design is 28 times faster than Bowtie2 running with 16 threads on dual Intel Xeon E5-2640 CPUs, and nine times faster than Soap3-dp running on an NVIDIA Tesla C2070 GPU.