Complex signal transform
A complex signal can be written as $s(t)=a(t)+jb(t)$, which indeed corresponds to two real signals ($a(t)$ and $b(t)$) "embedded" in the single signal $s(t)$. What makes this interesting is that $a(t)$ and $b(t)$ can be easily recovered from $s(t)$. This is true whether you write the complex signal in rectangular or polar coordinates.

Now, regarding how to interpret the phase of a signal. Let's say that the Fourier transform of $s(t)$ is $S(f)=|S(f)|e^{j\phi(f)}$. Imagine that you have an infinitely narrow band-pass filter with center frequency $f_0$. Then, you connect the signal $s(t)$ to the filter's input and see the output in an oscilloscope. You'll see a sinusoidal wave, with frequency $f_0$, amplitude $|S(f_0)|$ and phase $\phi(f_0)$. Here, "phase" indicates the difference, in radians, between a zero-phase cosine $\cos(2\pi f_0t)$ and the sinusoid you see in the scope.

In that sense, the phase $\phi(f_0)$ can be thought of as a "delay"; the sinusoid in the scope is just a cosine with a delay, $\cos(2\pi f_0t+\phi(f_0))$.

As an application, consider a signal that goes through an LTI system. The system doesn't touch $|S(f)|$ but it modifies the phase $\phi(t)$ -- that is, it changes the phase of each individual sinusoid. If this results in the same delay for each and every sinusoid, then the system output will be a delayed version of $s(t)$. If some of the sinusoids suffer different delays, then the signal at the system's output will be a distorted version of $s(t)$.