How to determine the aircraft tail number on a particular flight?
The step-to-step guide:

1. Search for Flight Number:


Get hold of the flight number, such as LH123 or AA789 for instance. This is usually found on your ticket, baggage tag or some email.
Enter it on a site such as www.flightradar24.com or www.planefinder.net.
Note down the registration (such as N1234A or D-ABYT) for the flight on the day your are interested in.


2. Search for this Registration:


Try entering it on for instance www.planespotters.net which is often considered a reliable source for aircraft data.
You should be able to get the aircraft history:


First flight and delivery dates (and hence aircraft age)
Previous operators and registrations. 
MSN (manufacturer serial number)