Moshe is of course completely right, but let me be a bit less abstract for a while and list a few examples that cover most of the contexts where the term "zero mode" is used.
When one discusses the motion of an extended object, such as the string, the position of a point along the string - the point is parameterized by $\sigma$ - is $X^\mu(\sigma)$. The "zero mode" of $X$ is then nothing else than the average over the string - the center-of-mass coordinate.
The dual quantity is the "zero mode" of the momentum - the integral of $P_\mu(\sigma)$: that's nothing else than the total momentum. These two zero modes, the center-of-mass position and the total momentum, are the quantities we would normally associate with a point-like particle. And indeed, in some approximation, the string may be considered to be a particle, and the zero modes are the usual properties of the particle.
If one solves differential equations on compact manifolds, such as the Calabi-Yau manifolds, one may find zero modes, too. For example, we may have a Dirac equation
$$D^\mu \gamma_\mu \psi(x_1, \dots, x_6) = 0$$
The solutions $\psi$ to this massless Dirac equations are just zero modes. The equation above only included functions of the compact coordinates. But if you consider both 6 compact and the 3+1 non-compact coordinates, one may consider spinors $\psi$ whose dependence on the 6 compact coordinates is given by the "zero mode" - the solution of the equation above. Such a field will then behave as a massless particle in 4 dimensions - it will solve the 3+1-dimensional equation with a vanishing mass.
In both cases above, there was an operator - either $-d^2/d\sigma^2$, or the Dirac operator $D^\mu \gamma_\mu$, that annihilated the solution and that's why the solution was ultimately called a "zero mode". In this context, it's useful to mention what are the "non-zero modes" or "normal modes". They are eigenvalues of the same operators with different eigenvalues.
For example, the operator $-d^2/d\sigma^2$ on the string may act on functions such as $\exp(in\sigma)$ and it will produce $n^2$ - the same with sines and cosines. These non-zero modes are eigenstates of an operator, so if you include all possible eigenvalues and the right degeneracy, you may reconstruct any function - by Fourier expansions.
The same is true for the Dirac example. All the spinors $\psi$ on the 6-dimensional manifold may be written as a combination of the "modes", some of which are the zero modes but most of them are non-zero modes. When one discusses supersymmetry, the non-zero modes typically come in pairs and it can be proved, but the zero modes - massless particles in 3+1 dimensions, as I mentioned - may come non-paired. So the number of zero modes of the Dirac equation (more precisely the number of solutions minus the number of "anti-solutions" with some opposite charge or chirality etc.) is a special number known as the "index". It is invariant under all continuous transformations. In heterotic string theory, it is interpreted as the number of generations (minus the number of antigenerations) of leptons and quarks we obtain in 3+1 dimensions.
We may continue with solutions to Dirac and similar equations. But consider a different background, e.g. an instanton in gauge theory. And include the gauge field term to the Dirac operator. Then this total Dirac operator will again have some zero modes on $R^{3,1}$; the zero mode will be nonzero and nontrivial especially in the region where the instanton is located. Any fermionic particles in 3+1 dimensions that has a zero mode will actually appear as a factor in a complicated product of many fermionic fields - and this multi-product interaction term is actually induced by the instanton.
The zero mode is a $c$-number-valued solutions of the right differential equation with the right operator. But its real meaning is that the whole quantum field should be expanded into modes - zero modes and non-zero modes. Each term in the sum has a $c$-number valued function of the "space" or "spacetime" and it is multiplied by an operator corresponding to the mode. The operators that multiply the zero modes in the expansion play a special role.
Instead of the Dirac operator, you may consider the differential operators for other fields, including bosonic fields such as the gauge field itself. The "zero modes" of the differential equation for the bosonic fields - essentially various forms of (covariantized) wave equations - describe deformations of the background. They're "collective degrees of freedom" of the instanton. It's called the 't Hooft interaction.
Note that the description "collective degree of freedom" applies to the string as well. The center-of-mass position and the total momentum are "collective degrees of freedom" for all the string "bits" along the string; they share these collective degrees of freedom. Some of the zero modes tell you how the whole object may move in space - like in the example of the string. But the fermionic zero modes can also tell you how the whole object may move in the superspace. And the zero modes of the instanton also include some modes whose variation determines changes of the size (and/or shape) of the instanton - or any other object.
This text was only meant for the reader to have some more concrete ideas about the words by Moshe which are of course totally valid.
Best wishes
Lubos
