It is called Semelparity.
Semelparity and iteroparity refer to the reproductive strategy of an organism. A species is considered semelparous if it is characterized by a single reproductive episode before death, and iteroparous if it is characterized by multiple reproductive cycles over the course of its lifetime.
You see semelparity as suicidal but it is more interesting to generalize survival after reproduction in all kind of species and ask why would an individual survive if it won't reproduce anymore anyway?
Why would an individual survive if it won't reproduce anymore?
From an evolutionary point of view, there is no point to keep living after the reproductive age. You will note a few interesting exceptions to the rule though, including Homo sapiens, where the female keep living relatively long after her reproductive age. Of course, modern medicine is part of the explanation but we tend to think that modern medicine alone is not sufficient. Another explanation is based on kin selection and is called the grandmother hypothesis. In consequence, semelparous species die right after reproduction.
Evolution of semelparity/ieroparity
I don't think this point can be summarized in a few words. In consequence, I will just invite you to have a look at wikipedia for an introduction on the subject.
How common is semelparity?
Iteroparity is more common. However it would be wrong to think semelparity is extremely rare or that it exists only in salmon species (in plural).
In invertebrates, Many insects (butterflies, cicadas, and mayflies) are semelparous as well as some arachnids and moluscs (some squids and octopus). In vertebrates, semelparity is rarer, but there are some frogs (i.e. gladiator frog), fishes (smelt, capelin), lizards (i.e. Labord's chameleon) and marsupials (such as the famous opossum). Finally there are semelparous plants (some agave and some bamboos). In unicellular species (and some multicellular species, such as some trees for example), the concept of semelparity/iteroparity may well not apply easily and depend on a philosophical discussion of the identity of an individual.