Laplace domain can be used to represent transfer functions so the phase is an absolute lag relative to the input phase.  This is a function of the transform Integral of x(t)je^-st.  There is a similar term in the fourier transform - rememember the e^-it type terms are an euler representation of cosines & sines which represent oscillations with a phase delay.  Its this term in each Fourier/Laplace that the phase information comes from.

In signal processing and then making analysis we take absolute values, relative values. Mat Lab considers absolute values. So, as our friend Aparna Murthy said "to check the minimal-phase and in turn helping to characterise the system invariance. 

The Fourier Transform measures the similarity between your signal x(t) and a kernel function exp(-j*2*pi*f*t) = cos(2*pi*f*t) - j*sin(2*pi*f*t). The similarity (or the amount of energy at that frequency) is measured by the energy of the coefficients for the cosine part (say 'a', and the sine part, 'b' (energy=sqrt(a^2+b^2)). The phase of your signal will be represented by the ratio between the coefficient for cosine and coefficient for sine, or better atan(b/a). Fourier analysis allows one to examine the function from another 'point of view', in another domain, the transform domain. The Fourier representation of a signal x(t) contains exactly the same information as that of your original function x(t). I do hope this helps :-)

As FSS indicates - if you are uncomfortable with the idea of phase, then instead just think of the relative values (sign and magnitude) of sin and cos components at a given frequency.

Also, another tip for your studies ... I struggled with the Laplace transform as a student, until I understood its connection with the Fourier transform. Loosely speaking, the former (with a real and imaginary axis) is a 2D generalization of the latter (imaginary axis only).  The imaginary axis captures periodic (oscillatory) behaviour, the real axis captures exponential growth/decay.