Potato Bags are widely available, with plenty of room for home-grown potatoes. Any dark container like this would work, but make sure there are holes for drainage or your potatoes will rot. This one comes with a flap to help harvest the final crop, but in all honesty, dumping the bag is much easier, you'll find it fills up fast. You can grab a few or test the potatoes for readiness using the flap. Make sure it's well sealed (this one pictured has velcro) before you add your soil. Pictured here are Fingerlings and Blue Russians. Use good quality seed potatoes and ensure each piece has at least two eyes with root development. The potatoes can be cut into pieces or planted whole if they're small.
If using a bag, fold it down so it's around 12" high. It makes it easier to work with and allows for more sunlight to get into the bag. Add a few inches of good potting soil, place your potatoes in the bag and cover with more soil. In a few days, the stems will appear and as they start to grow, add more soil - but don't cover the leafy top part of the stems, just the stems themselves (they'll grow potato shoots along the entire stem and eventually fill up the bag with potatoes). You'll do this 2-3 times as the potato plant grows until the bag is back to it's full height. These in the picture were planted May 21.
After 2-4 weeks, the potato eyes will start to sprout. Tall, leafy stems will develop. Add enough soil to cover the stems, but not the top leaves and roll up the bag to the top of the new soil level. You will do this a couple of times before the bags are full. The stems beneath the soil will send out new shoots that will eventually fill the bag with potatoes. Keep them well watered - potatoes like plenty of water and as with all containers, the bags can dry out quickly when the hot weather arrives.
The plants will start to sprout stunning purple or white flowers, depending on the potato variety. Once the flowers fade and the stems start to brown and topple over, the potatoes are ready to harvest.
To add a little colour to the garden, here's another option for planting potatoes.