Need of $\tt abs()$ method when plotting a power spectral density for a given dataset
Complex sines, or cisoids, $e^{iw}$ are fundamental functions for the study of linear  systems, even if the latter are real. And the FFT is a fast algorithm for a discretized version of the continuous Fourier transform.

So when you apply an FFT, it provides you with a bunch of coefficients $c_k$, which can be positive, negative, or complex. We are often interested in the energy carried by these coefficients, defined as their squared modulus, which can be computed  "complex number times conjugate equals square of modulus", or $c_k\times \overline{c_k} =|c_k|^2 $. This works with real numbers too. So instead of squaring up, you could as well multiply the FFT by its complex conjugate, pointwise. 

Calling it "absolute value" is a bit of an abuse. This term is generally understood for real numbers only, and "modulus" would be a better term. However, abs is a very common term in this situation. 

Note that sometimes, people are interested in the argument of the complex coefficients, and instead of 2D plots use the third dimension to "unwrap" the phase part, sometimes called phasor/polar representations: