Choose firm green marrows, ideally no more than a foot long, and with no yellow or brown patches.
Stored in a dark, cool, airy place, marrows can last for months, although their skin may toughen and turn yellow.
Marrows are courges in French and zucchi in Italian: hence courgettes and zucchini, both names for baby marrows.
I fear that the reason that marrow has become so popular in this country since its introduction in the eighteenth century has more to do with its propensity to grow to a huge size than its eating qualities. No other country deigns to eat marrow, but then no other country values size above flavour. One exception is the United States, but they have the pumpkin, which can be grown even larger.