blast is a method for automatic classical first-order reasoning, based on a tableaux prover. apply blast attempts a thorough search using sets of intro and elim rules. Like simp, rules can be annotated with [intro] and [elim] annotations, or they can be passed locally to blast (i.e. apply (blast intro: nandI1 nandI2 elim: nandE))
The methods fast, slow, and best use a different classical proof search engine than blast, each with a different proof search strategy. fast performs a quick, depth-bounded search, and is generally faster. slow is a more exhaustive search and more backtracking, and best performs a best-first search strategy using a heuristic based on the size of the goal. In general, if fast or blast can solve your goal, it will be the most efficient at solving it, whereas slow and best are only used if no other automated technique works.