A Brief Description
Other than the fish, the term salmon may refer to humans on bicycles riding against the flow of traffic. Like salmon, they have a certain season; they swim upstream; and, with any luck, they die. Sadly the last is more reliably true of actual salmon, who are more valuable members of society.
A Small Disclaimer
Most of this is from the perspective of a US citizen; your traffic culture and law may, and almost certainly will, vary wildly elsewhere. If I don't trip over myself making note of that every other sentence, it's only because I don't want to think about it too much, and end up weeping noisily, curled up in the corner, at how much better many countries are in these respects.
Riding against traffic is, firstly, illegal under every vehicle code I've ever seen or heard of. Furthermore, it is dangerous; riding against traffic places you outside of the mental expectations of other traffic. Of course, anyone operating a vehicle, no matter how large or small, should be alert and aware of everything going on around them. That's still no defense against motorized traffic which isn't expecting you to be there. I fully expect my gravestone to read "He had right of way" - that doesn't mean that I want to bump the date up any.
Indeed, the statistical evidence I've seen fully support this; Wachtel and Lewiston's Risk Factors for Bicycle-Motor Vehicle Collisions at Intersections shows the accident rate increasing by a factor of 3.6 for wrong-way traffic (and even higher in certain age groups.)
Now, don't get me wrong: I'm entirely in favor of dumb people getting themselves horribly injured or killed by their own fault. The problem is, when this sort of behavior becomes widespread, and there's no enforcement of the law, it can reinforce certain misconceptions. Local law almost invariably states that bicycle traffic is vehicular traffic, with the same rights - and responsibilities - as anything motorized. Still, many people honestly don't believe that, on both sides of the issue.
When large numbers of cyclists consistently break the law without repercussions, this reinforces the idea that bikes are toys, not traffic, encouraging a climate of disrespect for the law-abiding cyclist, who's been lumped in with the rest. The dangerous and illegal practice of riding against traffic clearly demonstrates a lack of regard for, or understanding of, practical cycling. Indeed, it will often be other cyclists most endangered by the salmon. They will often be traveling in the same general areas, causing conflicts when two cyclists approach each other. Even if the salmon travels on the right, a cyclist moving left to turn or pass can still be endangered. For that matter, a motorist trying to avoid making a little smoked salmon spread could swerve into a law-abiding cyclist.
While I've made my self-interest quite plain, there's also a great deal of societal benefit to encouraging safe, legal cycling. Healthier people, reduced traffic and fuel consumption, and an increased emphasis on local attractions and shopping over hopping in the car and onto the interstate are good for everyone. Salmon aren't just putting themselves and me in danger, they're doing harm, if perhaps only a small amount, to our entire culture.
The most common claims I've heard for wrong-way riding are safety and convenience. Both statistical evidence and reason demonstrate that safety for the rider is in fact sharply decreased. Similarly, riding against traffic can be more dangerous for your fellow cyclists, as described above. I see no possible argument for safety here, other than a possible false sense of security.
As for convenience, there are only a rare handful of cases where this may make sense. Most of the time, alternate routes will be available. For small stretches, dismounting and walking should not be a major issue. When there are no alternate routes available, the distance is great, and there's specific danger to riding with traffic, then this may be an excuse. This is rare for most folks, however - think of a highway under construction, whose right shoulder is being used as a lane for motorized traffic. Taking the left shoulder instead does make sense here. For city riding, there is almost invariably a better option.
In summary, salmon are people who either haven't thought things through or simply don't care, and endanger themselves and others, with no real benefit to themselves or anyone else.