The wavefunction is the basic object of quantum mechanics. The wavefunction for a system contains all the information about the quantum state that the system is in and gives a complete description of that part of the world at one particular instant. As the system evolves over time, the wavefunction also changes, so it can be written as a function of time Ψ(t). To solve QM problems in practice one usually factorises the wavefunction into a time-independent part ψ and a function of time T(t).
In the Schrödinger formulation of quantum mechanics, the wavefunction ψ(x) is a function over configuration space, in other words it takes a definite numerical value at each point or coordinate value in space. It obeys the time-independent Schrodinger equation.
The wavefunction is complex; the probability distribution of a particle over space is given by the absolute value squared, P(x) = |ψ(x)|2 = ψ(x)* × ψ(x).
Longer version with the interesting stuff
The time-dependent wavefunction Ψ(x,t) obeying the time-dependent Schrodinger equation, which describes the total history of a system. For a system with a time-independent potential of interaction, Ψ(x,t) can be separated into a linear superposition of energy eigenstates that factorise as Ψ(x,t) = ψ(x)T(t), where T(t) = e-iEt/h-bar and E is the total energy. One can also define Fourier-transformed wavefunctions that are functions of momentum ψ'(p) -- or wavefunctions that are functions of two or more coordinates, if one is describing a multi-particle system.
Physics implications of the wavefunction
Since QM is supposed to describe the behaviour of subatomic particles, the formalism tells us something unexpected: a particle may have a wavefunction that extends over all of space! This is utterly different from the classical picture (by which I mean the way people thought about things before QM came along) in which particles have definite positions and interact by action at a distance (for example Newtonian gravitation and electromagnetism).
Hence, quantum mechanics is non-local, which shows up in such things as the Einstein-Podolsky-Rosen paradox thought experiment and its physical realization by Alain Aspect, Bell inequalities and quantum teleportation. In the case of two or more particles, there may be quantum entanglement as a consequence of the fact that the wavefunction may not be factorizable into separate wavefunctions for each particle.
The wavefunction is complex number-valued. This is necessary for the consistency of the theory but somewhat confusing, since there are no complex numbers in measurable physical reality. The accepted solution is that the wavefunction itself is not observable! Instead, the product of ψ(x) and its complex conjugate ψ*(x) is interpreted as the probability density P(x) at the point x. In other words the probability that a single particle will be found in the small interval between x and x + dx is
dP(x) = ψ*(x)ψ(x) dx.
To make physical sense, we must impose the condition that the total probability that the particle be found somewhere be 1: so the integral of P(x) dx over all x must be equal to 1. This can be achieved just by multiplying ψ(x) by a complex number to normalize it, but only if ψ(x) is square integrable. Not every complex function over space can be a wavefunction: those that don't die away sufficiently fast as you approach spatial infinity can't be normalised.
Any observable physical property of the particle is represented by an operator O(x) which can be deployed as follows: one operates with O on the wavefunction and then multiplied by psi*(x). Then integrating over all space we obtain the expectation value of the operator
<O> = int ψ*(x) O(x) ψ(x) dx.
Provided that ψ is correctly normalized, this is just the average value one would obtain over many measurements of the same state ψ. For example the momentum operator in one dimension is p = -i h-bar d/dx, thus the expectation value of p is the integral of -i h-bar ψ* d ψ/d x.
Mathematical nature of wavefunctions
One important implication is that ψ(x) and eiθ ψ(x) describe exactly the same physics, where eiθ is a constant complex phase. If one allows wavefunctions that are not normalised, one can multiply ψ(x) by any complex number and it will still describe the same physical state. (Then the expectation values have to be divided by the norm int |ψ|2 dx.) Technically, this means that the space of wavefunctions is a complex projective ray space or Hilbert space.
Particular forms of the wavefunction are useful in some simple physical situations. The top hat function
ψ(x) = 1/sqrt(L) (-L/2 < x < L/2),
ψ(x) = 0 elsewhere
can be taken as a crude representation of the aftermath of a measurement of the particle position with a precision L/2.
ψ(x) = 1/sqrt(a(2π)1/4 exp(ip0x/h-bar - (x-x0)2/a2)
represents a wave packet (or wavepacket) with central position x0, position uncertainty Δx = a and central momentum p0. The momentum uncertainty Δp turns out to be h-bar/2a = h-bar/2Δx, saturating the Heisenberg bound.
Neither the top hat nor the Gaussian has a definite momentum or energy, so the form of these wavefunctions will not remain the same over time evolution and one would need to return to the full time-dependent Schrodinger's equation for the full solution.
In contrast, we can look at the plane wave
ψ(x) = K eipx/h-bar
which has momentum p and energy p2/2m. However, it is not square integrable! This stems from the fact that a particle with definite energy has infinitely uncertain position and so is in effect smeared infinitely thinly over the entire Universe. In practice, the plane wave is used to describe particle beams with a certain number of particles n per unit volume, so its normalization is different: we require |K|^2 V = n V, thus K = sqrt(n), where V is any volume of space that we integrate the probability density over.
Finally we look at the standing wave
ψ(x) = K' sin(kx)
which is a superposition of two plane waves of opposite momentum h-bar k and -h-bar k. It also has a definite energy E = (h-bar k)2/2m. The interesting thing here is that psi vanishes at x = 0 and x = m π/k for integer m. This makes it a possible description of a particle confined by an infinite potential well if we take the well to extend from x = 0 to x = m π/k and set the wavefunction to zero outside.