What are the differences between these three?
Why do they send you three different identifying codes?
The eTicket number (which is exactly the same as a paper ticket number) is your actual travel document's identifier. The ticket number is determined by the airline that tickets your trip (not the travel agent) and is used to track finances through the various airline reporting agencies. The first three digits identifies the airline that issued the ticket (which is not always the airline you are flying on).
The Itinerary Number, as Burhan mentioned, is simply a tracking number for the specific reservation system you booked through, be it an OTA (online travel agency), a bricks & mortar agency, tour company, etc. It is really only applicable when talking with the agency/agent that did your bookings. The GDS (global distribution systems - travel networking systems used by travel agencies) also issue their own Itinerary Number or Booking ID.
The Booking Reference, also know as a PNR (Passenger Name Record) or Record Locator, is the airline's internal identifier for your flight booking within their computer system. It is generated by the airline's computer system, not the travel agent or GDS. If your flights involve different airlines, there is often separate PNR's for each carrier for use within their respective systems, but you may be given only the ticketing airlines PNR (unless you ask specifically for all). When speaking directly with the airline you will be flying on, you will need to know the PNR for that specific airline.
As is always the case, there are always exceptions, as some airlines (such as major players in alliances) share more data with each other allowing them more flexibility with accessing reservations (such as referencing by another carrier's eTicket, etc)
An e-ticket is one that is issued to you by an airline (or travel agent) and this is just like a paper ticket - except instead of printing it at the travel agent, you can print it yourself. It usually has a bar code that is scanned by the checkin agent.
Depending on the airport and the airline, you can also use this ticket without printing it by simply scanning your mobile device and checking in.
The "e" is because it is issued electronically, rather than the old paper tickets:
Itinerary is simply your travel plan. It may include other forms of transport - for example, a bus ride from the airport to your hotel. Itineraries normally do not carry a reference; however I have seen some travel agents that provide packaged holidays provide references for an itinerary. This is simply done to save your preferences in their systems.
Synonyms for booking reference include PNR, Record Locator
A booking reference is a alphanumeric code that identifies your confirmed itinerary in the global reservation systems (these are called Global Distribution Systems, or GDS for short). There are 4 main ones - Galileo, Amadeus, Sabre and Worldspan. You really don't need to know which one you are on, since they all talk to each other.
Each travel agent or airline is plugged into these systems and the reference number generated is a unique identifier on this system for your booked itinerary.
A key difference to remember is that a booking reference is given for an itinerary; which have multiple travelers.
If you are traveling with three of your friends on the same booking (ie, you booked three seats together):
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