From The Miron Blog

While there is really no alternative that can match Ikea in terms of price, here are some options-- or last resorts, rather- for when you can no longer bear the sight of an Allen wrench. All of these stores conveniently have locations in the city, so you can see the furniture in person before making your order.

CB2; cb2.com
CB2 is a great alternative to Ikea; it's SoHo showroom [451 Broadway] is stocked with affordable, modern housewares and furniture for apartments. 

West Elm; westelm.com
West Elm lies somewhere along the shopping spectrum in between Ikea and Crate & Barrel, with higher quality furniture at a higher price. West Elm has a variety of locations in New York City, including two in Manhattan [112 W 18th Street and 1870 Broadway, at 62nd Street] and one in Brooklyn's DUMBO neighborhood [75 Front Street].

Room and Board; roomandboard.com
Room and Board receives rave reviews from it's customers, who praise the company for the quality and price of its products and for it's commitment to customer service. The store has a location 

Straight from the Crate; straightfromthecrate.com
Straight from the Crate, as the name implies, is a no-frills kind of furniture store, worth checking out if you are looking for basic lifestyle furnishings. Their showroom is located on the Upper East Side [1114 First Avenue, on the corner of 61st Street].

Downtown Furniture; downtownfurniturenyc.com
Not only does Downtown Furniture have low prices, but they also offer free shipping and assembly (take that, Ikea!) on all orders over $200. Downtown Furniture's main showroom is located in Manhattan [165 Grand Street, between Baxter and Centre], but they also have a second location in Williamsburg [686 Grand Street].