Why should we use t errors instead of normal errors? In this blog post by Andrew Gelman, there is the following passage:

The Bayesian models of 50 years ago seem hopelessly simple (except, of
  course, for simple problems), and I expect the Bayesian models of
  today will seem hopelessly simple, 50 years hence. (Just for a simple
  example: we should probably be routinely using t instead of normal
  errors just about everwhere, but we don’t yet do so, out of
  familiarity, habit, and mathematical convenience. These may be good
  reasons–in science as in politics, conservatism has many good
  arguments in its favor–but I think that ultimately as we become
  comfortable with more complicated models, we’ll move in that
  direction.)

Why should we "routinely be using t instead of normal errors just about everywhere"?
 A: Because, assuming normal errors is effectively the same as assuming that large errors do not occur!  The normal distribution has so light tails, that errors outside $\pm 3$ standard deviations have very low probability, errors outside of $\pm 6$ standard deviations are effectively impossible.  In practice, that assumption is seldom true.  When analyzing small, tidy datasets from well designed experiments, this might not matter much, if we do a good analysis of residuals.  With data of lesser quality, it might matter much more. 
When using likelihood-based (or bayesian) methods, the effect of this normality (as said above, effectively this is the "no large errors"-assumption!) is to make the inference very little robust.  The results of the analysis are too heavily influenced by the large errors!  This must be so, since assuming "no large errors" forces our methods to interpret the large errors as small errors, and that can only happen by moving the mean value parameter to make all the errors smaller.   One way to avoid that is to use so-called "robust methods", see  http://web.archive.org/web/20160611192739/http://www.stats.ox.ac.uk/pub/StatMeth/Robust.pdf
But Andrew Gelman will not go for this, since robust methods are usually presented in a highly non-bayesian way.  Using t-distributed errors in likelihood/bayesian models is a different way to obtain robust methods, as the $t$-distribution has heavier tails than the normal, so allows for a larger proportion of large errors.   The number of degrees of freedom parameter should be fixed in advance, not estimated from the data, since such estimation will destroy the robustness properties of the method  (*) (it is also a very difficult problem, the likelihood function for $\nu$, the number degrees of freedom, can be unbounded, leading to very inefficient (even inconsistent) estimators).
If, for instance, you think (are afraid) that as much as 1 in ten observations might be "large errors" (above 3 sd), then you could use a $t$-distribution with 2 degrees of freedom, increasing that number if the proportion of large errors is believed to be smaller.
I should note that what I have said above is for models with independent $t$-distributed errors.  There have also been proposals of multivariate $t$-distribution (which is not independent) as error distribution. That propsal is heavily criticized in the paper "The emperor's new clothes: a critique of the multivariate $t$ regression model" by T. S. Breusch, J. C. Robertson and A. H. Welsh, in Statistica Neerlandica (1997) Vol. 51, nr. 3, pp. 269-286, where they show that the multivariate $t$ error distribution is empirically indistinguishable from the normal. But that criticism do not affect the independent $t$ model.  
(*) One reference stating this is Venables & Ripley's MASS---Modern Applied Statistics with S (on page 110 in 4th edition).
A: It is not just a matter of "heavier tails" — there are plenty of distributions that are bell shaped and have heavy tails.
The T distribution is the posterior predictive of the Gaussian model.  If you make a Gaussian assumption, but have finite evidence, then the resulting model is necessarily making non-central scaled t-distributed predictions.  In the limit, as the amount of evidence you have goes to infinity, you end up with Gaussian predictions since the limit of the t distribution is Gaussian.
Why does this happen?  Because with a finite amount of evidence, there is uncertainty in the parameters of your model.  In the case of the Gaussian model, uncertainty in the mean would merely increase the variance (i.e., the posterior predictive of a Gaussian with known variance is still Gaussian).  But uncertainty about the variance is what causes the heavy tails.  If the model is trained with unlimited evidence, there is no longer any uncertainty in the variance (or the mean) and you can use your model to make Gaussian predictions.
This argument applies for a Gaussian model.  It also applies to a parameter that is inferred whose likelihoods are Gaussian.  Given finite data, the uncertainty about the parameter is t-distributed.  Wherever there are Normal assumptions (with unknown mean and variance), and finite data, there are t-distributed posterior predictives. 
There are similar posterior predictive distributions for all of the Bayesian models.  Gelman is suggesting that we should be using those.  His concerns would be mitigated by sufficient evidence.
