What does an electron's spin of 0.5 and minus 0.5 signify?
It is very tempting (and often also very useful!) to picture electron spin as an angular momentum vector, similar to a spinning top. Using this analogy, there are two properties (or numbers) of this angular momentum vector that we need in order to describe the electron spin. The first one is the spin itself and this is often designated the symbol $s$. The second number is the projection of the spin on a particular axis (often the $z$ axis) and this is assigned the symbol $m_s$.
You probably learned already that not all values of $s$ and $m_s$ are allowed in quantum mechanics. From measurements we know that an electron has a spin of $s=1/2$ (always positive) and its projections can be $m_s=+1/2$ or $m_s=-1/2$. The quantum numbers $s$ and $m_s$ are just numbers we use to label a particular state of the electron, but when we do measurements we also measure units. The units of angular momentum are J s (Joule second) and from comparing the classical definition of angular momentum with the rules of quantum mechanics, we can derive that one unit of angular momentum in quantum mechanics has a value of $\hbar$. This means that the angular momentum of the electron has a value which is basically half of what you would expect based on classical arguments and sometimes people like to emphasize the weirdness of this by saying things like &quot;an electron has to rotate twice around its axis to come back to its original position&quot;. Of course this not true because the angular momentum of the electron is not a consequence of a rotation at all (the electron is a point particle). In addition, the sign of the projection of $s$ has nothing to do with this hypothetical rotation either. The sign of $m_s$ does influence the interaction of the electron with an inhomogenous magnetic field. If you have a magnetic field gradient along the $z$ direction, electrons in one state move up while electrons in the other state move down and you will see two separated spots on a position sensitive detector behind the magnet. This is the basis of the famous Stern-Gerlach experiment. Note that you have to perform these kind of experiments with neutral systems that behave like a single electron because a charged electron would experience a much stronger interaction between its charge and the magnetic field (Lorentz force) than between its spin and the magnetic field (Zeeman interaction).
What I find very interesting about half-integer spin is that there is no orientation possible for which the electron spin does not interact with the magnetic field (which would correspond to $m_s=0$ or a perpendicular orientation of the electron spin with respect to the magnetic field).