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There are two stable Isotopes of nitrogen N14 and N15. The ratio of 15N/14N tends to increase with trophic level, such that herbivores have higher nitrogen isotope values than plants, and carnivores have higher nitrogen isotope values than herbivores. Depending on the tissue being examined, there tends to be an increase of 3-4‰ at each trophic level.

The tissues and hair of vegans therefore contain significantly lower percentage of 15N than the bodies of people who eat mostly meat. Isotopic analysis of hair is an important source of information for archaeologists, providing clues about the ancient diets.

https://en.wikipedia.org/wiki/Isotopic_signature#Nitrogen_isotopes

But how does N14 'disappear' from plant to animal?

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    $\begingroup$ Your last line does not seem to match the rest of the question as you did not say anything about N-14 'disappearing' anywhere else in the question. Yet, herbivores have more N-15 than plants and carnivores have more N-15 than herbivores because of bioaccumulation (en.wikipedia.org/wiki/Bioaccumulation). This claim is also supported by @Roland's answer. $\endgroup$ – another 'Homo sapien' Mar 6 '16 at 7:56
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Organisms accumulate certain isotopes because their metabolic enzymes have a slight preference for molecules containing that isotope. A well-known example is ribulose-1,5-bisphosphate carboxylase (Rubisco) which is responsible for CO$_2$ fixation in photosynthetic plants: this enzyme is somewhat more efficient with $^{12}$CO$_2$ than with $^{13}$CO$_2$, causing plants to accumulate $^{12}$C.

In animals, there is an enrichment for $^{15}$N because the deaminases and transaminases that transfer nitrogen from amino acids to urea (for excretion) tend to favor $^{14}$N. So in this case, $^{14}$N tends to be removed from the body faster than $^{15}$N when breaking down dietary protein. The differences are very small though: the relative enrichment of about 0.4% per trophic level corresponds to 0.367% $^{15}$N in a herbivore, compared to 0.366% $^{15}$N in a plant.

Source: this article.

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