Determinants are discussed (in a language relevant to this current question) in this MO question.
One place Picard categories naturally appear is as fundamental (aka Poincare) groupoids -- specifically those of infinite loop spaces (which are a refined homotopical version of abelian group objects in spaces, so form a natural source of abelian groups in categories. In fact Picard categories are equivalent to spectra which have only two consecutive homotopy groups, which up to shift we may as well take to be $\pi_0$ and $\pi_1$ -- one direction is given by the fundamental groupoid.
The important example of the Picard category of graded lines over a field arises this way from the algebraic K-theory spectrum of the field, via the determinant line construction (see eg Beilinson's paper referred to in the answers to the above MO link).
Another example important in rep theory is the Picard category of sheaves of twisted differential operators. This is discussed in detail in the famous "Proof of Jantzen Conjectures" paper of Beilinson-Bernstein.