What is it to be normal?

If I knew the answer to the title question, I probably wouldn't spend quite so much time at MO and MSE. (ba-dum ching!)

But seriously...

As a general rule, I imagine that the term arises for the simple reason that when you start studying a type of object (subgroup, topological space, field extensions, probability distributions), you quickly stumble upon the "good" set of such objects -- the ones that behave the way you want them to behave in order to set up a general theory. You then call these the "normal" such objects (because "good" sounds a little silly? But then you find "excellent" rings...) and build your theory from there up. Of course, a lot of the uses of the word are related to each other (as you indicate in a comment, there's a link between normal field extensions and normal subgroups), and certainly some uses of the word normal come from other sources. For example, the use of norms and normal vectors in linear algebra, according to the OED, probably date back to the 17th century use of the norm to reference right angles.

In any case, support for this explanation comes from the likewise extraordinary number of uses of terms related to "normal" (e.g., "simple", "regular", etc.), which are natural adjective to ascribe to basic objects if you're starting a theory from the ground up. It's entertaining to check out the sheer length of the PlanetMath encyclopedia entries beginning with N, R, and S due to the preponderance of terms that start with "normal", "regular," and "simple."

http://planetmath.org/encyclopedia/N/

http://planetmath.org/encyclopedia/R/

http://planetmath.org/encyclopedia/S/

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