the wavefunction
In classical mechanics a particle is a dot with a definite position and velocity. In quantum mechanics that dot dissolves into a spread-out complex-valued cloud called the wavefunction, written psi(x, t). It is the single object that carries everything the theory allows you to know about the system. Picture not a marble at a point but a ripple whose height (and phase) at every location tells you how likely the particle is to show up there.
Concretely, psi is a complex-valued function of configuration and time. Its physical content comes from the Born rule: the probability of finding the particle in a small interval dx around x is |psi(x, t)|^2 dx, so psi must be normalized, integral over all space of |psi|^2 dx = 1. Formally psi is an element of a Hilbert space of square-integrable functions (the space called L^2), and it evolves in time by the Schrodinger equation. In Dirac language psi(x) is just the position-basis component of the abstract state vector, psi(x) = <x|psi>.
The crucial honesty: the wavefunction is not itself observable. You never measure psi; you measure positions, energies, momenta, whose statistics psi predicts. Only |psi|^2 and the relative phases between parts of psi have physical consequence — a global phase e^{i alpha} multiplying the whole wavefunction changes nothing. And for N particles psi does not live in ordinary 3D space at all but in a 3N-dimensional configuration space, which is why the wavefunction is best regarded as a computational encoding of probabilities, not a literal wave sloshing in the room.
A free particle prepared as a Gaussian wave packet has psi peaked around one position; over time the packet spreads, so |psi|^2 broadens and the particle's location becomes less certain.
The wavefunction's squared magnitude is a probability density, and free packets inevitably spread.
The wavefunction is not a physical wave you could poke; it is a probability amplitude, and only |psi|^2 (with relative phases) is measurable, never psi itself or its overall phase.