How does a wave differ from a particle? I was asked this question by my teacher and didn't have a good response. My response included what a wave and particle were but not how they differed. So, how do they differ?
 A: They are both everyday visualizations of approximations to the behaviors we meet in physics. They are vague, everyday terms and are not meant to be rigorously defining. The idea that there even are rigorous definitions is wrong (or about as wrong as I am willing to pronounce anything in science), even if only in the sense that it is at least highly misleading.
The vague notion of particle can bear some or all of the connotations of:


*

*Characterization by a small number of state variables, that often pertain to a single location in space and time. Particles are not distributed over large regions;

*Indivisibility, or at least a transient or partial notion thereof. In particle physics, particles can be defined mathematically as irreducible representations of the Poincaré group. Don't worry if this is over your head: it roughly means that we carve our state space up into subspaces and if a physical system stays within a certain subspace even though all the transformations of physics (translations, rotations, boosts) might be applied to it, we call that system a "particle". This is not an indivisibility in space and time but rather in an abstract quantum state space, but the notion of indivisibility is very much at the fore.

*Quantum fields see particles as the result of a particularly simple "excitation operator" whose repeated application gives rise to a discrete ladder of states.


See my answer here for more details; hopefully you might be able to glean something of it. 
The vague notion of wave can bear some or all of the connotations of:


*

*A behavior that is the result of the interaction between a large, possibly infinite, number of low order, often identical, subsystems. A long linked array of masses on springs is a good prototypical example;

*The metaphor of the children's game of whisper down the lane, of a disturbance or state change being passed along a line (or higher dimensional analogue) of interacting systems;

*Waves are described by hyperbolic equations. From the Wikipedia article: 


The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to a fixed time coordinate, disturbances have a finite propagation speed. 

(2) and (3) are very alike, (2) being an evocative analogy for (3). (3) for me is the crucial property, and it is essential to notions of locality (no information propagation at faster than $c$) in modern physics. See my answer here for more details. For example, it is because the heat equation and mainstream heat transfer theory are not hyperbolic equations and do not give wavelike behaviors that we know that they must be approximations (albeit extremely effective ones) and cannot be compatible with special relativity.
A: I will differ with the chosen answer, by stating that one can define strict mathematical models for what a wave is and what a particle is in physics.
In Newtonian mechanics a particle has a uniquely defined  mass and volume and its motion can be described by translations of the center of mass system, and by rotations around an axis through the center of mass.
A wave is described by wave equations , i.e. sinusoidal solutions (varying in space and time) of differential equations. These waves are a transfer of energy in space over time and do not have mass or a center of mass. The classical waves on water and sound are an emergent phenomenon on ensembles of particles.
The classical electromagnetic waves are a phenomenon on sinusoidal changes in electric and magnetic fields transferring energy, and classically  ether was postulated as the medium  for electromagnetic waves. ( EM waves   can be shown to emerge from the underlying quantum mechanical level of photons which build up the classical wave, see this answer of mine, as ether was killed by the Michelson Morley experiment).
There was no problem in defining a particle or a wave in classical physics because waves were expected to be emergent from large ensembles of particles, not independent phenomena.
The confusion on the terms came with quantum mechanics where it was found that differential equations which describe wave behavior had to be used to describe/model  quantum mechanical particle interactions. Measuring the location of an elementary particle requires the solution of a quantum mechanical differential equation with wave behavior: the complex conjugate square of the wavefunction in quantum  mechanics gives the probability of finding a particle at (x,y,z) at time t, and this probability  has sinusoidal behavior.
The best example of the so called wave particle duality comes from the double slit experiment single electron at a time:


accumulation over time of single electrons

In (a), single electrons display the footprint of a classical particle, a point in (x,y) where z is the location of the screen.
The accumulation though of such footprints displays interference phenomena characteristic of waves. The accumulation is the probability distribution from solving the quantum mechanical equation "electron scattering off two slits" with given boundary conditions. This shows that the wave function must be sinusoidal. 
So classically a particle differs from a wave because a particle  has mass and a center of mass , and the wave is emergent from an accumulation of particles and does not have mass, only energy and momentum transfers on these underlying particles.
Quantum mechanically depending on the boundary conditions the accumulation of particles can show a wave nature in the distributions and the label, wave/particle  changes according to the boundary conditions. Somebody called them wavicles :).
