Quantum Mechanics: the bad news: our ordinary world is made out of weird, fuzzy, unpredictable stuff. The good news: the weird, fuzzy, unpredictable stuff is made out of unfamiliar but perfectly sensible math.
Related Pages: Physics, Decoherence, Many-worlds interpretation, Configuration space, Egan's Law, Timeless Physics
The biggest conceptual difference between the world of quantum mechanics and the physical world at the level we typically interact with is that it's much harder to specify the state of a system. Classical systems like a bowling ball or a planet have well-defined positions and velocity, and the state of such a system can be completely specified by just those two quantities. Quantities like position and velocity are called vectors, and in a 3-dimensional world a vector has component along each of the 3 dimensions. The state of a classical point particle can thus be given by just 6 numbers.
In quantum mechanics, particles don't have both a well-defined position and velocity, and as a consequence, the vector that describes a quantum system can't be expressed in just 3 dimensions. In general, there is no upper limit on the number of dimensions a quantum system can have, and so while the state of our bowling ball exists in two 3D spaces (one for position and one for velocity), a quantum system, in general, exists in a space that's similar to the 3D space we're used to, but with an infinite number of dimensions. This space is called Hilbert space. In order to be able to write down answers without using infinite numbers, quantum systems are usually mapped to other "spaces" like the 3D position and velocity spaces that we mentioned before. But information can be lost in this mapping, the same way a low-resolution photograph won't fully capture a 3-dimensional object. As a consequence of the lossy nature of this transformation, instead of the position of a quantum particle we instead get a distribution of possible positions. This is why quantum mechanics is often described as random or unpredictable.
Actually, quantum mechanics is perfectly predictable in Hilbert space. The only difficulty is that we don't live in Hilbert space, and while we have meter sticks and interferometers for making measurements in 3D space, we don't have any equipment for measuring Hilbert space directly. As a consequence, we have to make guesses about quantum systems based on what we see in 3D space. This is made even more difficult by the fact that once you measure a quantum system, it doesn't have the same distribution of possible positions it did before you measured it, so you can't sample repeatedly from the same distribution. Because we can't measure Hilbert space, the detailed dynamics of how exactly Hilbert space maps to real space, and what exactly happens in Hilbert space when you measure a system are still a matter of speculation and debate. People here have generally favored Hugh Everett's many-worlds interpretation over others.
In spite of that, quantum mechanics is a mature field, and even if there's some uncertainty about what the results might imply, actually doing quantum mechanics is not terribly difficult for single-particle systems. From a practical standpoint, the evolution of the state of the system (also called the wave function or the state vector) is governed by differential equations the same way it is in classical physics. In quantum mechanics, this equation is called Schrödinger equation and most of practical quantum mechanics is concerned with solving this equation for different sets of boundary conditions, and in trying to find a "space" in which the quantum system can be expressed and solved most easily. Systems of more than one particle are considerably trickier because the state vector has to be mapped into multiple different 3D spaces at the same time.