To my admittedly limited knowledge, aircraft for the majority fly on pre-determined tracks (NAT - North Atlantic Tracks). These are set by ATC each day and positions of said tracks are determined by the jetstream. which moves west-east over the atlantic. The pilots speak to Oceanic controllers at Shanwick (shannon and prestwick) when over the eastern n-atlantic and to Gander when in the west n-atlantic.
Due to the expanse of water there is no radar coverage and communications are done on a HF radio rather than the normal radio's. Pilots fly the track by entering waypoints into their fmc/mcdu like they do normal flight. The NAT waypoints consist of an entry and exit point, eg LIMRI and then lat/long points eg 56N020W. Crossing the atlantic pilots make position reports at every waypoint, or every 45mins (whichever is sooner). Should the controllers wish to contact the pilots they send an audible alert to the a/c using the a/c's SELCAL id. Each plane travelling over the atlantic must have a SELCAL, this means the pilots don't have to stay tuned in on the HF radio as this contains a lot of static noise.
I hope this is of some help to you, someone else maybe able to correct any errors I have made and/or give you more detail.