It's funny most international cities offer you some very inexpensive options to stay. American cities are a little behind on this front. New York City is behind as well. Where cities like Tokyo offer pods or places to rest for a few hrs or even small shared hostels, NYC pretty much only offers hotels. Now hostels are beginning to emerge in the city but not at hostel prices. These places usually are in the $100+ category which is still cheap by NYC standards but it's not Europe.
I have found the best options will be in Brooklyn or in New Jersey. Both are a very quick train ride to the city. If budget is your concern than this is your best choice. Another way to save money is to check in advance. NYC is one of those cities that is always sold out. Waiting till the last minute can sometimes pay off but that is rare so check in advance and decide if it's worth the wait.
Riff Chelsea it's got a great location and free wifi for $125/night
There are always cheaper options but you don't want to pay hourly or stay in a bad neighborhood. Check for deals too, as of this writing the DoubleTree in Midtown is $135/night 48% off. For more options check out FrugalTraveler's guide to cheap hotels in Manhattan.
Best Deal overall
The best deal also happens to be my favorite place to stay while in the city. Night is located a block off Time Square and 3 blocks from Port Authority bus terminal and a mere 10 blocks from Central Park. Now the one I usually stay at is on West 47th st off 7th known as the Time Square location. They do have another one two blocks south on West 45thst off 7th known as the Theater District. Rooms here can be found for $160/night and this hotel is super chic and modern boutique hotel.
If you're feeling Fancy
This is a tough category. Everyone has a different idea of what justifies spending $300, $500 or even $1,000 a night. Ordinarily I would have recommended the Waldorf-Astoria because of it's location, history and opulence but they recently closed for a 3+ year renovation. If I had to choose on location alone it's no contest, Marriott Time Square. The hollow atrium and the restaurants that jut out over Time Square are amazing. But beyond those location factors it's nothing special and given the location it can be difficult to get in and out.
That leaves me with my pick for a pricey night. 1Hotel Central Park. I know it's not the Plaza or the Ritz or even the 4 Seasons. I think these places over priced, they don't really offer more amenities at least not ones that are included in your rate. 1Hotel is gorgeous and has a great location. The design is filled with reclaimed wood and steel littered with glitz and high end finishes. Well worth the modest $400/night, oh and did I mention they are pet friendly?
If you prefer a hotel even more off the beaten path or more niche if you will Cup of Jo posted a great article with 5 great hotels.