Interpretations of Quantum Mechanics
It is important to note that I will not be going into any of the math behind quantum principles. I don't know it well enough to explain it to other people and I don't think it has much relevance to this discussion.
The Copenhagen Interpretation
This is the most popularly held view of the implications of quantum theory. The basic consensus arrived at by Neils Bohr and Werner Heisenberg is that the wave function is a description of the observers knowledge of the to-be-observed system. The wave function is not taken to be an objectively real thing necessarily, but merely a description of what can be known of the system. The quantum system can display wave like properties or particle like properties and whichever is solely dependent on what apparatus the observer chooses to observe the system. The quantum system is not said to be either a particle or a wave, only that it displays these properties under certain measurements. All systems are quantum systems, but the quantum effects of the system are dependent on Planck's Constant which is extremely small, and so are only apparent in systems whose size approach this number.
Ensemble Interpretation
The Ensemble Interpretation is basically the same as the Copenhagen Interpretation but says that the wave function only applies to groups of similar phenomena and not individual particles.
Many Worlds Interpretation
The many worlds interpretation assumes that the wave function is an objective description of reality and that the collapse of the function at the point of observation is an actual split of possibilities into parallel universes. Under this assumption, all possible outcomes of the wave function do happen, but the act of observation limits the observer to just one of the possible outcomes.
Alright so; none of these interpretations say anything about "random". The main insight that quantum mechanics offers about reality is that no observation can be objectively discrete from the observer. In other words, our perception of reality cannot be isolated from the way in which we observe it. To me, this seems intuitive but it is a departure from the commonly held scientific view that existed before these things were discovered. The typical objective and passive perspective that science would like to adopt is fundamentally untenable. One cannot observe a system without 'looking at it' and looking at something is in itself an effect. If you have read Kant, this is basically what he is talking about when he says that we cannot know "the thing in itself", we can only know things about it based on the way it and we interact with each other. The general rule in particle physics is to not talk about particles as discrete and objective entities since those cannot be perceived, but only as the relationships that they have with other systems.
The perceived randomness in quantum mechanics comes from the impossibility of knowing the conditions of a system without observing it, and the inevitable effect that observation will have on the system. Classical physics will assume certain properties of a system because these properties can be observed ahead of time, and that observation has a negligible effect on the behavior of the system. When dealing with quantum systems, this effect is not negligible and in fact can alter the system drastically. Because it is impossible to know what exact state the system is in before it is observed, it is impossible to objectively determine how it will behave once it has been observed.
However, the development of the wave function can be determined and is not random. This is why we are able to accurately predict the way a system will evolve without knowing the exact initial conditions.
In short, quantum mechanics recognizes the fundamental limitations of how we interact with reality. It doesn't say anything about what actually exists, or even whether anything objectively exists. It merely explains the way quantum interactions evolve. It recognizes that it is impossible to know a system without observing it, and the observation expands/alters the system that is being observed.
Implications for Determinism
Quantum mechanics essentially puts determinism squarely within the realm of metaphysics. Its not that it disproves determinism, it is that it disproves the assumption that it is possible to determine. In other words, quantum mechanics does not say that reality is fundamentally probablistic, only that by the very nature of interactions, it is impossible to know all the information contained within any system. There is no outside perspective that one can assume to see all aspects of a phenomenon because they are intrinsically a part of the phenomenon through the act of observing it.