How do you establish complete versus partial mediation in a simple mediational model? I recently received this question from a student:

In a simple mediation model, if I have found the indirect effect (ab)
  to be significant and the direct effect (c') to be small and
  insignificant, does that mean I have full mediation or partial
  mediation?


 A: Definitions
I'll use the $a, b, c, c'$ notation common to simple mediation, as shown here.
Assuming there is a positive effect to be mediated (i.e., $c > 0$) and any underlying causal arguments are satisfied then


*

*Partial mediation occurs when $0 < c' < c$.

*Complete mediation occurs when $c' = 0$.


Theoretical interest concerns the underlying parameters rather than the sample estimates of these parameters. 
Testing for partial mediation
Significance tests can be applied to test for partial mediation.
Significance tests can support inferences such as that $ab$ is significantly greater than zero, or that that $c'$ is significantly less than $c$. 
Testing for complete mediation
Significance tests can not be readily applied to the test of complete mediation. The fact that $c$ is significant, and $c'$ is not significant is insufficient to prove complete mediation. First, the difference between significant and non-significant is not necessarily significant. Second, even if the reduction is significant, a non-significant $c'$ does not prove that the value of $c'$ is zero.
I imagine there is discussion of this approaches to testing for complete mediation in the literature, but a few options spring to mind:


*

*Equivalence testing: You could test the null hypothesis that $c' < \hat{c}$, where $0 < \hat{c} < c$, and $\hat{c}$ is deemed to be sufficiently close to zero or sufficient less than $c$ that rejection of the null hypothesis is seen as an argument for complete mediation being plausible.

*Confidence intervals: You could get confidence intervals on $c'$.

*Bayesian approaches: You could use Bayesian approaches to get a posterior density on $c'$ and if the 95% credibility interval was sufficiently small, you might argue that the mediation is plausibly close to being complete. A quick search revealed this article (Bayesian mediation analysis).


General thoughts on reporting mediation analysis
It seems to be me than when quantifying the degree of mediation, both the percentage reduction of $c$ to $c'$ is interesting as well as the size of the indirect effect. The terms partial and complete mediation suggest a binary distinction that is probably rarely true in social science research applications.
Rather, reporting a mediation analysis should focus on quantifying the degree of mediation both in percentage terms and in terms of the size of the indirect effect. It should also quantify the uncertainty in these estimates. 
Review of David Kenny's points
As an additional point, it is worth noting that David A. Kenny acknowledges the issues related to significance testing for mediation on his webpage. I quote the main passage here:

Note that the steps are stated in terms of zero and nonzero
  coefficients, not in terms of statistical significance, as they were
  in Baron and Kenny (1986).  Because trivially small coefficients can
  be statistically significant with large sample sizes and very large
  coefficients can be nonsignificant with small sample sizes, the steps
  should not be defined in terms of statistical significance. 
  Statistical significance is informative, but other information should
  be part of statistical decision making.  For instance, consider the
  case in which path a is large and b is zero.  In this case, c = c'. 
  It is very possible that the statistical test of c' is not significant
  (due to the collinearity between X and M), whereas c is statistically
  significant. It would then appear that there is complete mediation
  when in fact there is no mediation at all.

A: The Baron & Kenny approach is somewhat outdated - nowadays it is recommended to use a bootstrapping approach to test for mediation (Preacher & Hayes, 2004). One problem with the B&K approach is, that it is possible to observe a change from a significant $X\rightarrow Y$ path to a nonsignificant $X\rightarrow Y$ path with a very small change in the absolute size of the coefficient. 
A more direct test of mediation is to test the difference of $c - c'$ (which, in most cases, is equivalent to testing the indirect effect $ab$). The bootstrapping approach has much more statistical power and does not rely on multivariate normality assumptions (which are violated in indirect effects anyway).
To directly answer your question:

Q: In a simple mediation model, if I have found the indirect effect (ab) to be significant and the direct effect (c') to be small and insignificant, does that mean I have full mediation or partial mediation?
A: According to B&K: full mediation. According to P&H: not necessarily full mediation.

Preacher, K. J., & Hayes, A. F. (2004). SPSS and SAS procedures for estimating indirect effects in simple mediation models. Behavior Research Methods, Instruments, & Computers, 36, 717-731. doi:10.3758/BF03206553
