What is a nuclear wave packet? What is the definition of a nuclear wave packet ?  I often see the term used but i don't know how it is defined. It seems to be connected to the Born-Oppenheimer approximation. Is it defined there and is it only a valid concept within the Born-Oppenheimer approximation or is it a more general concept that also holds outside of the BO-approximation ?
 A: In quantum chemistry, "nuclear wave packet" is generally just a synonym for the wavefunction (or sometimes the probability density) of the nucleus or nuclei. This concept is most valid in the Born-Oppenheimer approximation, when the nuclear and electronic wavefunctions are treated as factorizable, i.e. $\Psi(\vec{r},\vec{R}) = \phi(\vec{r},\vec{R})\chi(\vec{R})$, where $\Psi$ is the overall wavefunction, $\phi$ is the electronic wavefunction, $\chi$ is the nuclear wavefunction, $\vec{r}$ represents the spatial coordinates of the electrons, and $\vec{R}$ represents the spatial coordinates of the nuclei. 
As Ben points out in his answer, this usage of "wave packet" may be an abuse of terminology, since "wave packet" typically suggests a propagating wave, whereas the probability density of the nuclei in atoms or molecules in stationary states are time-independent, but I believe this usage is standard. (See for example Fig. 6 in this review: J. H. Posthumus, "The dynamics of small molecules in intense laser fields," Reports on Progress in Physics, in which the $v=4$ vibrational wavefunction of $H_2^+$ is referred to as the "nuclear wavepacket.")
In the context of the Born-Oppenheimer approximation, the most significant aspects of the nuclear wave packet are that (1) the positions of the nuclei are essentially independent of the positions of the electrons (leading to the separability of the wavefunction described above) and (2) the nuclear wavefunction is highly localized compared to that of the electrons - i.e. the nucleus is small and compact, while the electronic wavefunction is spread out and diffuse. (Think of the de Broglie wavelength, $\lambda = h/mv$, which implies that the length scale characterizing the spatial extent of a particle is smaller for heavier particles.) The separability and narrowness of the nuclear wavefunction allow for certain simplifying assumptions when solving the Schrodinger equation, as outlined in these useful lecture notes on the Born-Oppenheimer approximation from NYU. These assumptions lead to the ability to divide the Schrodinger equation into a nuclear part and an electronic part which can be solved mostly separately: the position of the nucleus is independent of the electrons, while the electron wavefunctions can be treated as adiabatically following the position of the much heavier nucleus.
A: The term "wave packet" usually refers to a wave train that propagates. As a simple example, a free electron can have some wavefunction $\Psi(x,t)$. Because the Schrodinger equation is dispersive, the shape of the wave tends to change with time, becoming more spread out.
In the Born-Oppenheimer approximation, this would refer to the wavefunction $\Psi(x,t)$, where $x$ is the position of the nucleus. If the molecule is vibrating, then $\Psi$ could actually have a nontrivial dependence on $t$. However, I would think that in most cases you would be interested in eigenstates of energy, which are standing waves, and therefore there would not be any wave packet propagating through space (e.g., no probability current).
