Given some data $y$, dependent on parameter $\theta$, I have some density $p(\theta | y)$. I now want to infer what value of $\theta$ is most `likely' to have originated $y$. One possibility of doing this is calculating the MAP estimate of $\theta$. However, I was wondering if there are other ways of finding candidates for $\theta$, preferably in a Bayesian setting?
The choice of estimator really depends on how you plan to use it afterwards. Each of the common estimators are associated with a particular loss function which specifies what you are interested in capturing. For example, the posterior mean minimizes the mean squared error (MSE), the wikipedia link mentioned by Procrastinator (in the comments) gives a more thorough discussion.