Ferries from Norddeich on the mainland run roughly hourly in peak season and take about an hour. Norddeich is served by train from Emden and Cologne (via Norddeich Mole station, which actually runs to the ferry terminal). The ferries are run by AG Reederei Norden-Frisia; book ahead in summer.
Cars are allowed on the ferry and in the town area of Norderney, but bicycles are the better option once you arrive — most of the island is car-free anyway, and parking is limited and expensive. A bike is more useful than a car for exploring.
There is also a small airport (Norderney Airport) on the island, primarily used by light aircraft from the mainland and private charters.