My main confusions are with the symbols used, and what they mean, my textbook for my math course was designed by fresh grad students, and it doesn't really identify what all that stuff mean.Letfbe a function defined on an open interval containingc(except possibly atc) and letLbe a real number.

means that

for each real ε > 0 there exists a real δ > 0 such that for allxwith 0 < |x−c| < δ, we have |f(x) −L| < ε. or, symbolically,