What are real-valued and complex signals and why is the Fourier transform of a real-valued signal Hermitian?
When one analyses data, outcomes both depends on the nature of the data, and their handling (eg a transformation, like the Fourier transform). One can consider binary data, categorical data, data in two dimensions, etc. To develop a theory and the related tools, it is useful to define where the d ata dwell.

A real signal is simply a signal that takes values that are real. The signal can be discrete or continuous, mono- or mulitdimensional, the observed values are real. The set of real values $\mathbb{R}$ is quite interesting. Since it is a continuous field, it allow a lot of "signal-like" mathematical operations: addition, product, convolution, and a lot of properties related to limits. $\mathbb{R}$ is one of the simplest mathematical structure allowing such operations. 

But a natural tool to study signals, spectra,  is related to Fourier analysis. And Fourier uses complex arguments. Why shall we use complex transforms to study real signals? That is a complicated story, but let us say this is mandatory, to cope with linear and time-invariant systems. See for instance: Why cosine is not an eigen signal?

So one can also use complex-valued data. The complex field $\mathbb{C}$ is, maybe, the second simplest mathematical structure allowing the same operations as $\mathbb{R}$, without pain. And a real signal is just a subset of complex-valued data: take a complex object, whose imaginary part of zero, and you have a real signal.

For the last question: real signals are just a subset of complex signals. So when you analyze them (the reals) with a fully complex transformation, you can expect that the outcome will be restricted too. One of the restriction is that real signals have certain symmetries in the Fourier domain, that fully complex signals don't have