In terms of fare collection, most commuter lines I know (NJT, LIRR, Metro-North, and SEPTA) have you purchase a ticket from one station to another and then present it to a ticket collector on the train (who usually puts a "seatcheck" on your seat, which indicates where you're going, so you don't go further than you paid for).
Transit systems vary a little bit. The NYC subway and NY/NJ PATH system are a single fare that you pay when entering the system. You can ride that train as far as you want, and depending on the setup you may be able to transfer to another line for free. The same fare applies whether you ride one stop or the whole line. On the Washington, DC Metro, fares change depending on how far you go. You put your ticket through a machine when you enter the system (and it records your starting point) and then again when leaving, and the appropriate fare is collected.
Somewhere between commuter and transit is light rail (like the River LINE and Hudson-Bergen Light Rail in NJ). On those, you purchase a ticket (with no specified destination) and before boarding a train, you have it stamped with a date/time when it expires (long enough to ride the whole system). On board, the tickets are subject to spot checks by fare inspectors, who make you have a ticket and that it's valid.
Train design varies from system to system. Generally transit systems use multiple-unit cars, where each car (or pair of cars, in some cases) provides power for itself, usually running on an electrified third-rail.
Some commuter railroads operate the same kind of trains (Metro-North and LIRR use third-rail (though the cars are generally, while SEPTA and NJT (and some Metro-North cars) operate from overhead catenary. SEPTA, NJT, Metro-North, LIRR, and MARC also use locomotive-hauled cars for some trips (or all, in some cases). The cars get their power for lighting and such from the locomotive, which can be powered from overhead lines (NJT/SEPTA/Metro-North), third-rail (Metro-North/LIRR), or a diesel engine (NJT/MARC/LIRR/Metro-North).
RiverLINE light rail cars are diesel powered, while Hudson-Bergen Light Rail cars draw power from overhead lines.