A train ticket is a voucher issued by a railway operator that enables the bearer to travel on the operator's network. Tickets can authorize the bearer to travel a set itinerary at a specific time (common for long-distance railroads), a set itinerary at any time (common for commuter railroads), a set itinerary at multiple times, or an arbitrary itinerary at specific times. (The last two categories are often called passes: the former is often sold as a discounted block of trips for commuters; the latter is often sold to vacationers -- for instance, European Eurail passes.)
Tickets are conventionally printed on paper and are collected by a train conductor during the journey. Often, the conductor will give the passenger a seat check -- another voucher indicating how far the passenger may travel on the system. Some systems (America's Amtrak, for instance) have two-part tickets that permit the passenger to retain a cancelled ticket stub; others (the North American New Jersey Transit and MBTA commuter rail systems, for instance) do not.
Seat checks are changed frequently to ensure that passengers cannot retain and reuse them from journey to journey. (Conductors typically collect checks before stops to prevent this.)