The precise details of how party lines were set up varied from place to place, since there were quite a number of different systems. It was also the case that independent phone companies often used different arrangements from the Bell System.
What all systems had in common was the fact that two or more people shared a telephone line back to the central office (telephone exchange). When any one party was using the phone, all the other numbers would be busy too, e.g. if party A was on the line then anyone trying to call party B would get a busy signal even though party B wasn't actually using his phone. As you've already realized, it also meant that if any one party on the shared line was engaged on a call, anybody else picking up on that same party line could eavesdrop and join in the conversation.
The main differences between the systems were in the number of parties per line and the arrangements for ringing the called party. The basic 2-way party line generally had what was called fully selective ringing (as Bob described above), which means that your phone would only ever ring for your calls. If you picked up when the other party was using the phone you would overhear his conversation, but you would not know when his phone was ringing.
4-way party lines could also be arranged for fully selective ringing, but partially selective ringing was also employed with coded ringing, typically either one long ring or two short rings. So your phone would ring for both your calls and for calls to one of the other four parties on the line. You just had to remember whether "your" ring was one long or two short. The other two parties on the 4-way line would similarly have one long or two short rings, but you would never hear them.
The coded ringing principle was extended to allow for 8 and 10-way party lines, using different combinations of ring cadence: long, short-short, long-short, short-short-short, etc., although a few places used tuned ringers and different frequency ring voltages to provide fully selective ringing on such large party lines.
Although much rarer, there were even 20-way party lines in some very rural areas!
As for conflicts between users, they could happen if one person was noted for particularly long, chatty calls, or where somebody liked to eavesdrop to get all the local gossip. It was often in the terms of service that calls should not exceed a certain length without a break of a minimum period to allow other parties use of the line. Many jurisdictions actually had laws requiring a person to give up the line if another party on the line declared that it was needed for an emergency call.
The party line was essential to the plot of the 1959 movie "Pillow Talk," in which Doris Day shared a line with Rock Hudson. You'll often find similar references to party lines in other movies of the era, such as the running gag in the 1962 film "Mr. Hobbs Takes A Vacation."
Party lines were used in many other parts of the world. Huge multi-way party lines weren't that common here in the U.K. due to the much more compact nature of the country, but the basic 2-way party line was once very common here, and survived well into the 1980s.