My preferred way of thinking about it is to think mainly in terms of comparisons. Suppose that I have two acids, HA and HB. If HA is a stronger acid than HB, then B- is a stronger base than A- is. I agree with the implication of the previous comments, that the language can mislead the unwary. However, we can flesh out Petrucci's comment, as Borek has suggested. Suppose we think about phosphate monoanion, which has a pK of about 7.2 at low ionic strength. It is not a strong acid, in the sense that would tend to ionize very little in water. Its conjugate base, the phosphate dianion, would also not tend to ionize much in water; in other words, it is a weak base.