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There are a couple of ways of dealing with this, and snapping is involved in all of them. You can snap to points, end points, vertices and edges. If you have created polygon X and want to digitize polygon Y so that the two share one common edge, the best tool right to use is the Auto Complete Polygon tool. You should be able to look that up in the help system. Other options depending on which versions of arcGIS you have available would involve building a topology, and using the topology editing tools such as 'construct features' but that starts to get a bit involved depending on what your needs are.