This may go waaaaay above and beyond your question, but here goes:

The ARC ticket stock itself doesn't really have anything to do with e-ticketing any more than printing an email on company stationery makes the email any less electronic than printing it on plain copy paper. Some agent terminals (especially shared-use terminals or stations operated by a contractor--MCO, where the ground staff is AA, comes to mind) are equipped only with the more standardized ticket stock printers (since the thermal paper and printers each airline uses for its own receipts is unique to that airline). This only affects the kind of paper the boarding pass is printed on, but the physical paper in your hand is still just a regular boarding pass tied to your electronically ticketed reservation. You'll know because it clearly says "boarding pass" on it. The boarding pass can be lost and reprinted because it is simply a pointer to your electronic record.

In some instances, though, the airline will actually print the electronic ticket to a paper ticket on ARC ticket stock. Once this happens, the ticket no longer exists in electronic form, and the physical paper you possess carries intrinsic value--it's like cash. You can actually take that ticket to any airline counter or perhaps even any travel agent and exchange it for another ticket (minus any change fees, of course). When you go to board a flight, you must turn over the physical ticket to "pay" for that flight.

If you lose that piece of paper, it's as if you've lost cash (the value may be recoverable through a long and arduous process where the airline has to verify that the physical ticket hasn't been exchanged and its value used by someone else before they will reissue the value to you, since it can be exchanged without an electronic trail (the PNR, or electronic reservation record, can be canceled without voiding the ticket, since the ticket retains its intrinsic value).

Now, it's rare to get a paper ticket. Airlines don't like dealing with paper tickets not only because of the additional cost of the paper stock itself but also the additional accounting logistics required, as the paper ticket must be physically handled and reconciled with the flight records. However, in some cases, a paper ticket may be required, such as on certain (increasingly rare) itineraries involving connecting carriers (especially international itineraries involving smaller regional carriers). However, most paper tickets you'll get at check-in are due to a mismatch between the PNR (Passenger name record, the electronic reservation record) and VCR (Virtual Coupon Record, the electronic flight coupon book that replaces the paper ticket flight coupons)--in other words, the flights in your reservation don't match the tickets to pay for them.

With enough of the right keystrokes, an agent should be able to resync your PNR and VCR by either revalidating it (simple and fast; only possible if the routing and fare class haven't changed) or reissuing it (more complicated; necessary if something else has changed), since your boarding pass can't be issued if there's a mismatch.

However, in a crushed airport environment--especially during IRROPS and when door closure is imminent--an agent rebooking you will sometimes take one of two shortcuts: 1) either rebook you in full Y (much simpler than trying to rebook you in your initially-booked fare class, since the agent doesn't have to figure out what class you were initially booked in and try to convince the system to book you into that class when it might not be open for sale; the ticket will then be reissued and the PNR and VCR will match, allowing them to print your boarding pass) or 2) rebook your flight in your PNR without exchanging the ticket and then print your ticket to paper, thus removing the VCR, making it possible to print your boarding pass without the VCR and PNR being synced (since there is no VCR). The ticket and the reservation won't match exactly, but the airline's accounting department can then reconcile the two together later when they're not under pressure to get you on your way or get a flight's door closed.

You'll know that you're actually receiving a paper ticket because the airport agent will print out two items on ARC ticket stock: the boarding pass, which will say "FLIGHT COUPON REQUIRED," and a physical ticket (usually stapled behind the boarding pass), which says "FLIGHT COUPON" at the top middle.

There's some interesting reading in a few other places that might help shed some light on this:
http://www.flyertalk.com/forum/alask...l#post12195977 (start here and read forward, especially posts #15-18)
http://www.flyertalk.com/forum/ameri...ml#post9255270
http://www.amadeus.com/qa/documents/...0-%20Mar07.pdf (Amadeus instead of Sabre, but the concepts are similar)
http://answers.yahoo.com/question/in...5105517AADe32m (wow, the first useful thing I've ever seen on Yahoo! Answers! )
http://www.aa.com/i18n/agency/Bookin..._ticketing.jsp (inside look at e-tickets from AA's/Sabre's perspective)
http://en.wikipedia.org/wiki/Electro...th_an_e-ticket, http://en.wikipedia.org/wiki/Passenger_Name_Record, http://en.wikipedia.org/wiki/Record_locator, http://en.wikipedia.org/wiki/Boarding_pass, http://en.wikipedia.org/wiki/Airline...tion_.28PNR.29 (a couple of other random Wikipedia articles)