The original (and most fundamental) definition of a ghost field was one for which, when it is quantized, the particle-like excitations do not have positive norm. One of the first places this came up was in attempts to canonically quantize the electromagnetic field in a relativistically-invariant way. If you look at the (Feynman-gauge*) propagator for the electromagnetic field,
$$\langle0|T\left[A_{\mu}(x)A_{\nu}(y)\right]|0\rangle=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{-ig_{\mu\nu}}{q^{2}+i\varepsilon}e^{-iq\cdot(x-y)},$$
this looks just like four massless Klein-Gordon propagators, along with an overall factor of $-g_{\mu\nu}$, which is $+1$ for $\mu=\nu$ spacelike, but for $\mu=\nu=0$, there is an overall factor of $-1$. This can be traced back to that fact that, if we expand $A_{0}$ in creation and annihilation operators, the timelike polarization state $|k,\hat{0}\rangle=a^{(0)}_{k}|0\rangle$ created by a creation operator $a^{(0)}_{k}$ in the mode expansion of $A_{0}$ as negative norm, $\langle0|a^{(0)\dagger}_{k}a^{(0)}_{k}|0\rangle<0.$ These ghosts appear in the Gupta-Bleuler quantization method, and demonstrating that the negative norms states are never produced physically is a necessary part of the formalism.
By choosing a different basis of polarization states for the quantization of the electromagnetic field, it is possible to get different kinds of ghosts. If (when $\vec{q}$ is along the $z$-direction), instead of the timelike polarization vector $\epsilon_{(0)}^{\mu}=\hat{0}=[1,0,0,0]$ and the longitudinal spacelike vector $\epsilon_{(3)}^{\mu}=[0,0,0,1]$, we use a basis containing
$$\epsilon_{(\pm)}^{\mu}=\frac{1}{\sqrt{2}}[1,0,0,\pm1],$$
we find that states created with these polarizations have zero norm. The zero-norm ghosts are in many ways easier to deal with; in particular, these lightlike polarization vectors can be used to show how the effects of the timelike $\epsilon_{(0)}^{\mu}$ and longitudinal $\epsilon_{(3)}^{\mu}$ cancel out in such a way as to make the overall time evolution unitary.
The other examples in which ghosts arise also involve the creation of excitations with negative or zero norm. This is simplest to see with the Fadeev-Popov ghosts used in quantization of gauge theories. The ghost fields in this case are Lorentz scalars but are quantized according to Fermi-Dirac statistics, with the sign differences resulting in the creation of negative-norm states.
For a scalar field theory with a Lagrangian density like
$${\cal L}=-\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4},$$
the negative sign in the first term is a problem. If the states are to have positive energy when the theory is quantized with this ${\cal L}$, then it is necessary for the $\phi$ creation operators to, once again, create states with negative norm. If the creation operators are the usual ones, then the unconventional negative sign in front of the kinetic term will lead to negative energies.
However, in a theory like this, there is a trade-off possible. Since the equations of motion derived from ${\cal L}$ and another Lagrange density
$${\cal L}'=-{\cal L}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi+\frac{1}{2}m^{2}\phi^{2}+\frac{\lambda}{4!}\phi^{4}$$
are exactly the same, we can use this Lagrangian instead. Now the particle states will have positive kinetic energies when the theory is quantized according to the usual method, but the potential energy is now unbounded below, since we have switched $V(\phi)\rightarrow-V(\phi)$. The negative norm states have been traded for an instability; which formulation is more useful to use will depend on the circumstances.
*With care, this can be generalized to other gauges, with $-i(g_{\mu\nu}-\alpha k_{\mu}k_{\nu}/k^{2})$ in the numerator of the propagator.
