Probabilistic algorithm with two-sided error
Monte Carlo methods are inherently not one-sided, though it's dubious whether they're usually two-sided. The general idea is that in order to estimate the expectation of a random variable $X$, we take the average of many samples of $X$.

For example, suppose that you're a petroleum company, and you're trying to decide where to look for oil. For every square in some grid you have some random variable $X_i$, expressing the probability that there is oil in square $i$, that you can estimate using your (noisy) data. You have some threshold $\theta$ in mind, and will start drilling in every square $i$ for which $X_i \geq \theta$. Using a Monte Carlo algorithm (say, simulating the geological history of the area), you can determine $X_i$ up to some error $\epsilon$. This translates to a two-sided error algorithm.

You can also think of computation in general as having two-sided error. Hardware errors occur with extremely low probability, and they can affect the answer in every which way.