Is my time series stationary? I am using R and have found that both KPSS ( Kwiatkowski-Phillips-Schmidt-Shin ) and the adf (Dickey-Fuller) tests indicate stationarity, having a p-value of 0.01.
Here is a plot of the original data:

However, when I plot the correlogram, it looks as though the data are non-stationary.

So what do you think? Is the data stationary or non-stationary? 
 A: Tests for stationarity are notorious for having weak power so keep that in mind. As mentioned in the comments, it helps to use judgement as well. A weakly stationary process by definition has a constant mean and variance. 
While your correlogram (which I'm assuming is the autocorrelation function) shows significant autocorrelation, this does not necessarily mean the series is non-stationary - its telling us that the observations are not independent.
By just looking at your plot, the series does look stationary but highly seasonal. 
Another diagnostic you can try, which is pretty much analogous to the ADF test, is to fit an AR(1) model to the data. If the AR(1) coefficient estimate is (significantly) less than 1, then we have evidence of a stationary process. If the AR(1) coefficient is approximately 1, the process is more likely to contain a unit-root and is non-stationary.
A: I agree with all the points regarding seasonality and visual inspection above.
One additional thing: ADF tests the null of a unit root whereas KPSS tests the null of a stationary process. So if both have a p-value of .01, you actually have conflicting results (which may happen, tests being capable of type I and II errors, of course) at conventionel levels.
A: The declining wave pattern in the ACF and the ACF(1) > .9 indicates nonstationarity.  If shown, the PACF would reveal a spike at PACF(1).
Transform the data to (1 - B), then examine the ACF and PACF.  If a statistically significant spike occurs at PACF(12), then the series may be seasonally non-stationary, thus requiring the series to be transformed (1 - B) (1 - B)^12 prior to modeling the residuals.
Seasonal non-stationarity is a bit tricky to identify. The guiding rule is do NOT over difference in an attempt to achieve stationarity. 
