I just booked my first Amtrak ticket ever for a trip in mid-September, and the email I got says I have "1 reserved coach seat," but I don't see a seat number or anything to give me a unique seat anywhere. When, if ever, will that happen? Do I get emailed an assignment as the trip gets closer, do I get one when I present my ticket at the gate, etc.?
The distinction between a "reserved coach seat" and an "unreserved coach seat" is that they (sort of) guarantee that there will be seats for all passengers with "reserved" seats but not for passengers with "unreserved" coach seats. From the Amtrak website (bolding mostly mine):
While reserved coach seating is available on our long distance routes and many trains that travel short / medium distance routes, the train cars are different depending on where a train travels, so some of the amenities are different as well. ... Advance reservations (by train, not by individual seat) are required.*
Only a few short-distance trains have unreserved coach seating, where tickets are valid on any train unless restricted by the fare paid.
While seating is not guaranteed, unreserved coach seats have most of the features that reserved coach seats have...
The trains that have unreserved coach seats are Auburn (CA)–Sacramento–San Jose; Milwaukee–Chicago; Philadelphia—Harrisburg; and San Luis Obispo–Santa Barbara–LA–San Diego. I would guess that these lines are often used by commuters who want a little more flexibility in their train times.
As to how the boarding experience works: You get on the train, find a seat, and stow your luggage. Occasionally, the conductors will direct you to particular cars depending on your destination as you walk up to the train on the platform. Once the train is underway again, the conductors will go through the cars and ask all new passengers to show their tickets.
*Editorial note, May 2019: when I wrote this answer in 2015, the Amtrak guidelines at at the time followed this sentence with, "Once you make your reservation, a seat is guaranteed." This sentence is no longer on the current version of the page, which strikes me as mildly suspicious.
Unlike European railroads and most airlines, Amtrak never issues specific seat reservations, on any train. Even if riding in the Acela's pricey first class, it is first-come, first served. Generally, this is less of a problem for the solo traveler, but it can be a major complication for the couple or family that understandably wants to sit together. The nearest exception I can think of is in the case of sleeping accommodations, in which one reserves a specific room. The reasons Amtrak give for this policy don't withstand analysis, and given that the option of specific reservations at extra cost are a revenue source for European carriers, they make even less sense. Certainly, the needless queuing and mad rush for seats at many train stations diminishes the appeal of rail travel in this country.