When estimating parameters with MAP, why is it written that we are estimating the "mode"? I thought it would be the mean of the posterior distribution?
In maximum a posteriori estimation (MAP), we use the maximum of the posterior distribution to derive point estimates of whatever we are interested in. The maximum of a distribution is also called its "mode" (assuming a unimodal distribution).
This is not the mean of the posterior (or any other) distribution. The difference is relevant when the posterior is asymmetric.
When you are calculating the MAP estimate of the parameter, you assume posterior quantity to be of unimodal distribution like normal distribution or chi-square distribution. And the mode of the unimodal distribution doesn't mean the most frequent value, it refers to the local maximum or the peak point. That's is why you call MAP as the mode of the distribution.