Most root crops require too much growing depth to successfully grow indoors, however two notable exceptions are radishes and round varieties of carrots. Both can easily be grow indoors all year round in pots. Just make sure there is enough depth of compost for them to grow down.
Potatoes are another crop which can be grown well indoors, providing you have the space. I think the best way to do this is to plant your potatoes in plastic sacks. Start off the potatoes in a sack which is only one third full of soil/compost. Roll down the sides of the sack so the light can get in, but once the potatoes start growing, roll up the sides a bit and and more soil etc. Keep doing this until the bag is almost full.
Mushrooms can be grown in a draught free dark place with a temperature of between 50-60ºF. The easiest way to grow mushrooms is to purchase a bag of mushroom compost which is already full of mushroom spores. Follow the instructions and you could be eating your home grown mushrooms withing 2-3 weeks.
Dwarf varieties of peas and beans can be grown indoors in pots on a windowsill. They also have beautiful flowers and they make interesting/decorative houseplants. Pick the pods when they are young to encourage prolonged cropping.
There are obviously other herbs and vegetables you can grow indoors, but these 6 are not so commonly grown at home indoors and I thought it would be worth sharing the information.