Most people know that trout and salmon are the same family (salmonid). Char, grayling and whitefish also belong to the salmonid family.
One of the things that distinguishes fish in the salmonid family from other fish is that salmonids (trout, salmon, char, grayling and whitefish) need clean, cool water and a healthy habitat to live in. This why invasive species and plants are such a big deal when found in these environments - they threaten the population of salmonids.
Of the salmonids, trout and salmon are very closely related and most trout anglers enjoy catching salmon and vice versa.
Just as there are many different species of trout (rainbow, brown, brook, lake, apache, gila, steelhead, etc.), there are different species of salmon (coho, chinook, Atlantic, sockeye, and chum).
And to the untrained eye, it can sometimes be difficult identifying your catch (especially since these fish have a lot of similar characteristics).
The easiest way to to tell the difference between salmon and trout (when in doubt) is to count the number of rays on the anal fin.
All trout have 12 or less rays in this fin, whereas salmon have 13 or more rays.
This rule is an interesting bit of trivia, it can be useful in helping you identify the difference between trout and salmon.