A "farmers' market" is a collection of produce and locally-created goods (usually edible) sold by those that produced them, directly to the shoppers. They are usually a weekly occurance.
When shopping at a farmers' market, you get the opportunity to meet the grower of your food, rather than getting only as close as peeling a sticker off of your apple. You can support local businesses and say a big "no thanks" to far-away produce that damages the ecosystem by hauling in the goods via fossil fuels. The food itself is healthier, as it is fresh and didn't have such a distance to travel. By shopping at a local market, you also get the variety of the year and you get to sample the seasonal goods year-round. This enables you to get "back to your roots" and eat as your ancestors did: different produce at different times of the year.
It's also pretty fun to get into the routine of grabbing your fresh groceries on a weekly basis. You'll start to make connections and you'll be "rooting" yourself into the community.
Also note: the goods are usually cheaper than at the store.
Farmers markets are held at various points in the city.