Ok this isn't too hard to explain.
There are only 2 light rail lines in North Jersey (there is one in South Jersey that connects Camden to Trenton) and also there is the Newark Subway, but I doubt that you will be using that.
Both light rail lines cost $1.75 per ride.
The new light rail in Newark runs from NJ transit's broad st station down McCarter to Penn Station Newark. This line does not go to NYC, but you could take a train from Penn Station that does (however this would likely be self-defeating as the trains leaving from Broad Street Station also go into NYC).
The other light rail line, and the larger and more frequently used one, is the Hudson-Bergen light rail which runs from up in Union City I believe (although this will certainly be extended north) down to Bayonne (and there have been rumors about extending it south to Staten Island, but this is somewhat unlikely to happen). The Hudson Bergen line does not go into NYC, but however it connects with the PATH which will take you into NYC at Exchange Place in Jersey City, Pavonia/Newport, and Hoboken.
The PATH is a Subway system that is run separately from NYC's subway. It runs between New Jersey and Manhattan. You can take it all the way from Newark into either the World Trade Center Station (in lower Manhattan) or to Midtown next to Penn Station. The PATH stops in Harrison, Jersey City, Hoboken, and 4 stops in NYC before reaching PENN Station. The PATH is pretty nice and will be coming out with new train cars within a year or two. It costs $1.50 per ride, however if you put $10/$20 or more onto a metro card the ride only ends up costing $1.25 each way.
NJ Transit is the main heavy rail commuter line in NJ. The price of a ticket is obviously more expensive on NJ Transit than on PATH or the Light rail lines. Take a look here:
http://www.trainsnj.com/ This will show you where all of the NJ Transit stations are. There are too many to go into depth about where they are all located. The main difference between NJ transit trains is that some go to Hoboken where you have to switch to the PATH or take a Ferry into NYC and then some go directly into Penn Station without having to get off the train at Hoboken. The trains that go directly into Penn Station are called Midtown Direct.
If something is still unclear you may want to check out the web pages of NJ Transit, PATH, or the light rail. They all have web pages where you can find detailed schedules and time tables.
You might, however, want to look into the Buses and Ferries depending on where you are living. Some buses end up getting you into the city in a shorter period of time, for a cheaper price. There are also fast ferries leaving from the Atlantic Highlands that go directly into NYC in 35 minutes. The fast ferries are, however, more expensive than the train that leaves from that area.