"Buckets" is a nickname/metaphor for "fare classes".

All airline tickets are not created equal: some cost more than others, and some have more restrictions than others (even if they're in the same cabin). The airline sells a certain amount of seats on each flight for a set combination of rules and prices, i.e. fare classes.

An FTer might be interested in booking flights in certain buckets to ensure that he can or will be upgraded, or will qualify for a mileage or status promotion. Or she might want to see how many seats of each type of fare have been sold to determine her chances for an upgrade or bump. Or if the price seems high, he might check how many seats in the higher buckets are left, to see if an airline might make more inventory available at the cheaper prices.