Critical elements to adjust during a model's training phase in a machine learning context are bias and variance. When dissecting prediction errors in predictive models, we can classify them into two groups: bias-induced errors, and variance-induced errors. The friction between these errors gives rise to the bias-variance trade-off phenomenon. Understanding prediction errors (bias and variance) is crucial when dealing with model predictions. The ability of a model to balance these errors effectively is a balancing act. A deep understanding of these errors can lead to the creation of more accurate models and can help prevent overfitting and underfitting errors.

Bias is the discrepancy between a model's predictions and the actual values, and signifies the inability of the model to fully recognize or learn from patterns in the training data. High levels of inaccuracy in both the training and testing data are usually the outcomes of models with high bias.

Variance, on the other hand, relates to the variability in model predictions for a given data point, representing the data's spread. Models with high variance error often focus excessively on training data, to the point of memorizing it and thus, fail in generalizing for data they’ve never encountered before. They perform well with training data but not with testing data.

The bias-variance trade-off is a cornerstone in a supervised ML algorithm's goal to predict the function (mapping) of the output variable (Y) from the input data (X). The mapping function, also referred to as the target function, is the function our supervised ML algorithm is trying to emulate.

Underfitting occurs when a model cannot comprehend the underlying data patterns, and these models usually have low variance but high bias. This may be due to a lack of sufficient data to build a fitting model, or when a linear model is attempted for nonlinear data.

Conversely, Overfitting transpires when our model perceives both the underlying pattern and the noise present in data. Typically, these models yield high variance and low bias, and are intricate and prone to overfitting.

Balancing between a model with low variance and high bias or high variance and low bias is key to avert overfitting or underfitting of the data. Building a good model requires a balanced compromise between bias and variance to minimize total errors, which never leads to underfitting or overfitting.

Bias simplifies the target function's estimation by using a model's simplifying presumptions while variance pertains to the changes in estimating a target function based on different training data. The conflict between these error types is recognized as a trade-off. Thorough knowledge of the bias-variance trade-off is key to understanding the behavior of prediction models.