A windowsill or two (best if it is southern facing), a pot that will fit on that windowsill (a few solo cups or milk jugs will do), potting soil, seeds, and water.
My kitchen windowsill is southern facing and is wide enough to place a decent sized pot. I like to repurpose gallon milk jugs. By cutting off the top ½ to 2/3 of the milk jug you will make a flat wide base. Wash out the bottom of the milk jug and you are ready to go.
Now that your pot is ready, add your potting soil. Be sure to fill to the top and tamp down to slightly compact the soil. Plant your seeds to the desired depth and water thoroughly until the soil is moist. I chose not to cut drainage holes in the bottom of the jug. If you do the same you will need to be sure to not overwater your plants as too much water can be harmful to your plants. You can put drainage holes in the bottom of the jug by placing several (5-7) knife slits into jug. You can add a plate or tray under your jug to catch any excess water that drains through.
There are many veggies you can plant in your windowsill this winter. I like to grow greens that will come back after repeated harvests. Things like lettuces, spinach, mustards, and kale can be planted densely and harvested multiple times. A normal harvest of one jug will yield enough lettuce to provide my family with 2-4 servings of lettuce. If you enjoy salad you will want to have multiple jugs in various stages of maturity. This will enable to have homegrown greens almost nightly throughout the winter.
Winter gardening outdoors in a northern climate can be difficult but not impossible. Tools to help your winter garden thrive can include: row cover, cold frames, hoop house, and/or greenhouse.
All of these options are bigger projects than your windowsill garden and can require quite a bit of time and/or money to start. To get the most bang for your buck I would suggest the cold frame. Some scrap wood for framing and an old window or two is all you need to build one and get started.
A cold frame is nothing more than a square or rectangular sized micro green house. Use your old windows to guide you with the sizing. The windows will need to cover the top of the box so the glass acts as a greenhouse inside the cold frame. Build the box to fit the windows and you will be in good shape. A hinge and a stick to prop up the window will make managing your cold frame easy.
Since your cold frame is actually outside and in the ground you can plant just about anything, root crops included: carrots, beets, greens, cabbage, even potatoes are fair game in your cold frame. On warm or sunny days you will have to check the temperature inside the cold frame. The temperature can rise quickly on sunny days so be sure to crack the window open as needed.
Regardless of which method you chose growing a garden in the winter will help brighten your day, enjoy homegrown food, and count down the days until spring.