The word salient originally meant leaping. It comes from the Latin word salire, which meant leap, and it is one of the many words that were invented by this blog's favourite essayist Sir Thomas Browne in his Pseudodoxia Epidemica:
Salient animalls, and such as move by leaping
But from these bestial beginnings, salient was then taken up by mathematicians, who decided to use it for a discontinuous leap in a graph. If the value of an equation changes suddenly then it leaps from one position on the graph to another. Thus is it salient.
From that you got the idea of anything that sticks out, and from there to today's meaning of important. However, many of salient's cousins are still jumping about. Salmon, for example, probably also derives from the Latin salire because salmon are always jumping out of the water. And so do the words assault (to jump at) insult (to jump in) and exult (to jump out). Moreover, there is a supersault, although we don't spell it that way anymore, because somewhere on the journey from Latin to English it got corrupted to sobersault and from there to somersault.
The Inky Fool inspecting his ceiling