This covers getting to NYC or to Jersey City/Hoboken on public transportation (or to Newark, for that matter!) 

Please note that the Airtrain from EWR will not operate between May 1, 2014 to July 15, 2014 due to maintenance.

But assuming that you're staying in Manhattan, the first thing you need to know is where your hotel is in relation to  the Manhattan street grid.  Here's a nice map you can use -- it's centered in "mid town" near Time Square but you can scroll around to other areas  -- locate your hotel so you can better judge the different services.  Unless you take a taxi or car service that will take you directly to your hotel, you're going to have to walk, take the subway  or a NYC taxi from where the train or bus lets you out

For taxis (including sample rates), van and car services, which take you right to your hotel, see this 

Note that the shared shuttle services generally do not get great reviews. 

If you want a car service, Dial7 and Carmel Limo generally get very good reviews but are, of course, the most expensive.  There are two ways to get limo service, both requireing an advance reservation.  The first is where you call them after you've collected your luggage and they send a car out to get you.   The second is where they park at the airport and the driver waits for you to come down into the baggage area -- these are the guys holding signs with people's names. The latter is the most convenient and expensive option, of course!
 

There are also 4 other public transportation options to reach NYC by train and bus, two by train and two by bus

To get to New York Penn Station located between 31/33rd streets and 7/8th Avenues ( map)

see Airtain

Please note that the Airtrain from EWR will not operate between May 1, 2014 to July 15, 2014 due to maintenance.

Take the Airtrain to the the Newark Airport train station. Then transfer to a NJ TRANSIT train to Penn Station NY, the last stop (this is all done on one ticket, not two).  Make sure you don't get out at Penn Station NEWARK.  Also, don't get a ticket for an Amtrak train because that will cost you a LOT more.  Trains run 7 days a week, 21 hours/day (no service between 2:00-5:00 AM) at least every 15 minutes and more frequently during the daytime.  The cost is $12.50 and total travel time including waiting for a NJ Transit train at the Airpot Tain Station should about 30-45 minutes, less during peak times when trains are extremely frequent.

Train service is the most reliable, the cheapest and often the fastest because you are not at the mercy of traffic.  It's the most highly recommended mode of travel from EWR to NYC by the NYC Destination Experts here on TA.  But you've got to schlepp your bags around and transfer trains. 

 

To get to Lower Manhattan (World Trade Center & Wall Street), Newark, Jersey City and Hoboken

see Airtrain

Though most hotels are in mid-town, some are downtown near the WTC and Wall Street (and, of course, some are in Jersey City, Newark and one is in Hoboken).  Take the Airtrain to the the "Newark Liberty Airport Train Station". Then transfer to a NJ TRANSIT train to Penn Station NEWARK (not New York).  Trains run 7 days a week, 21 hours/day (no service between 2:00-5:00 AM) at least every 15 minutes.  Do not get a ticket for an AMTRAK train because that is MUCH more expensive. Get off the train at Newark Penn Station (which has an enclosed walkway to the Newark Gateway Hilton) and go to a PATH train in the same station. PATH is the subway that runs between NJ and NY, costs $2.00 and is unrelated to the NYC subway system.   PATH will take you to lower Manhattan -- here's a PATH system map .  Note that PATH has three lines that connect, one to WTC, one to Hoboken and one to 33rd street (not too far from Penn Station). 

If you don't feel like schelpping your bags on the train and want to take a taxi, see above link for "taxis, van and car services"

 

To get to the following three locations by express bus service:

Grand Central Station (41st Street between Park & Lexington Avenues)  ( map)

Bryant Park (42nd Street & 5th Avenue)   ( map)

Port Authority Bus Terminal (41st Street between 8th and 9th Avenues)  ( map)

Use Newark Airporter Express bus service

Direct service to NYC from EWR (no transfers)  -- but you are at the mercy of traffic which can be significant.  Buses come to every stop ostensibly every 15 minutes during the day, 365 days a year (departures every 30 minutes before 6:45 AM and after 11:15 PM).  Remember that traffic affects these busses in both directions, not just inbound to NYC since in order to get to the airport every 15 minutes they can't be delayed outbound from NYC.

Reivews of the Express bus are somewhat mixed - see  Yelp reviews

If the above don't get you close enough to your hotel to walk , assume that you'll pick up a NYC taxi for the final leg of your trip but keep in mind that during rush hour or inclement weather taxis can be hard to come by