A cognitive bias is a psychological predisposition to error, where as formal and informal fallacies are logical errors arising from structural flaws in reasoning and generally arise from formal deficiencies or violations of principles, though they overlap to a certain extent. Informal fallacies usually rely on cognitive bias (think appeal to emotion, for instance).
WP article has a subsection on classification.
From T. Edward Damer's Attacking Faulty Reasoning, avoiding informal fallacies must satisfy three criteria: they must be acceptable (reasonably true), relevant to the conclusion, and provide good grounds for the inference. The author classifies this way:
- Fallacies of Linguistic Confusion
- Begging-the-Question Fallacies
- Unwarranted Assumption Fallacies
- Fallacies of Missing Evidence
- Causal Fallacies
- Fallacies of Irrelevance
- Irrelevant Appeals
- Fallacies of Diversion
- Fallacies of Deductive Inference
Note that highly relevant to these groups is the Toulmin Method for understanding argumentation as outlined in his work Uses of Argument where fact, warrant, backing, rebuttal, and conclusion are structural elements of argument and inference.
Also see the fallacy and bias recommendations in this SE post.