Ultimately, it is different for each person and it is whatever you want it to be. For the experienced angler, a Brown Trout or Smallmouth Bass under 18 inches may not raise an eyebrow but for a less experienced angler, that 12 inch Brown Trout is huge. For many, any native Brook Trout they catch is a trophy in and of itself. And everybody is getting excited about any Tiger Trout (Brown Trout x Brook Trout hybrid) that they catch. I remember the first trout I caught on a hook and worm and my first fly rod caught trout. They were certainly all trophies to me.
"Science", of course, as a bit to say about what a big fish is. But before we get there, let us acknowledge that it depends greatly upon the individual and where they are fishing. Here in Wisconsin's Driftless Area, while there are certainly large trout to be caught, your "average" stream trout is less than 12 inches in most places. For me, once a fish starts growing "shoulders", it moves from average to "nice", maybe even large.
First, a bit about expectations. We certainly don't catch the average fish in the stream - and that is generally a good thing. As you should expect, small fishes are much more common than larger and older fishes but we are generally targeting the older and larger fishes. The figure above is pretty illustrative of Brown Trout populations in a really good Driftless spring creek. Note the "Driftless medians" are the median values for the number of fish per mile of each size class and of all trout. The average stream in the region sampled (see: WDNR Southwest Region report) has a little under 350 trout per mile and 33 trout over 12 inches per mile. So that 16 inch Brown Trout may not be a trophy but in most Wisconsin streams, it is a bit of a rarity. Interestingly, if you want to catch a real trophy, generally your best bet are the more marginal trout waters that tend to get a little warm but provide plenty of small fishes for the big Brown Trout to eat.
A lot of what a trophy is depends upon where you are fishing. In Wisconsin, the most obvious difference is difference between inland waters and the Great Lakes and their tributaries. For example, the state record Brown Trout is 40.6 inches and 41.5 pounds whereas the inland record is less than half that weight (18 lb. 6 oz. and 34.3 inches). Even at that, the inland record comes from Lake Geneva, a large inland lake. Simply put, lake fishes tend to have to expend less energy and they live in places with more food that is more energy dense.
Then there is the Brook Trout anomaly. The Wisconsin Great Lakes record is but two ounces heavier than the inland record from "the Prairie River". Everyone knows that the Prairie River did not produce a Brook Trout an ounce short of 10 pounds. Local tales are that the fish was caught in a local spring pond and it makes a lot more sense than thinking it came from the Prairie.
When Wisconsin trout anglers were surveyed (source), they considered a 10 inch Brook Trout and a 12 inch Brown Trout to be quality fish and a 14 inch Brook Trout and 20 inch Brown Trout to be trophies. My experiences in Wisconsin pretty well mirror this though I probably would have said 9 inches for a quality Brook Trout and 14 inches for a quality Brown Trout but that probably reflects the fact that most of my fishing is in streams dominated by Brown Trout.
Length Categorization System
Fisheries biologists have taken the world records and developed classifications based on percentages of world record fish length as a way to assess lakes and streams with things like Proportional Stock Density, Relative Stock Density, and other metrics that really are not all the interesting nor important. The basic idea is that we can compare fisheries based on the proportion of different size classes. A fishery with a larger proportion of larger fish is generally thought to be better than one with a lot of smaller fishes and few larger ones.
What is maybe more interesting are the size classes developed to determine these ratios used by some fisheries biologists. Based on proportions of the world records for length; stock (~20%), quality (~36%), preferred (~50), memorable (~60), and trophy (~75%) are determined and widely applied. For Brown Trout, the stock length is 8", quality is 12", preferred is 16", memorable is 20", and trophy is 24". A trophy Brook Trout is 23 inches. Based on these categories, there are very few Brown Trout and almost no Brook Trout that are "trophies" in Wisconsin's streams. Now travel to the Nipigon River and Brook Trout trophies are relatively common.
For the warmwater anglers, a memorable musky is 42" and a trophy is 50", for walleye, they are memorable at 20" and trophies at 25", a memorable Largemouth Bass is 20" and a trophy 25", Smallmouth Bass are memorable at 17" and a trophy at 20", and Bluegill are memorable at 10" and 12 inchers are trophies. At least this is what some scientists had come up with. Your mileage may vary.
All this leads me back to the first paragraph - a trophy is what you want to to be and it is different for everyone. It will be different for each person at different times in their lives. The first two trout - a brook and a brown - that I caught on a hook and worm from a Nicolet National Forest stream and fried up in cornmeal were great trophies to a 12-year-old. The brood stock Rainbow Trout - one about 23 inches - I caught a few years into learning to fly fish are lesser trophies to me today but they sure made me feel accomplished at the time. To many, a trophy is what they would put on the wall. My walls are empty, not because I have not caught fish that could go there but the idea of a fish on my wall just doesn't excite me much.
What gets me excited now is catching fish the way I like to catch them.