While the train option outlined by mazeem is probably the best balance of time and cost for most people, there is a less expensive option for those who have more time. Namely, you can take the MTA subway from Jamaica to the World Trade Center (or from Howard Beach to Fulton Street), and then take the PATH to Hoboken. The fare will be $10.50: $5 for the AirTrain and $2.75 each for the subway and the PATH.
According to Google, the travel time for the LIRR itinerary is around an hour and 25 minutes, while for the subway itinerary, via Howard Beach and the A train, it's around 10 minutes longer, and via Jamaica and the E train, it's around an hour and three quarters.
I got the Howard Beach itinerary by choosing "fewest transfers" in the search options, and the Jamaica itinerary by choosing "less walking."
If you are traveling on the weekend, you can save yourself three dollars on the LIRR fare by buying CityTicket at the kiosk. This costs $4.25, as opposed to the $7.25 off-peak fare.
A side benefit of this approach is that if you take the A train to Fulton Street, you can walk through the recently opened Oculus:
This is especially useful in foul weather, as it means you can avoid going outside.