I think the handmaids were women who had socially transgressed, such as being lesbians or divorced. June was married to a divorced man so according to gilead they were adulterers.
Yes, that's what I thought.
If you've on your first marriage to a Commander, you get to be a Wife. If the pair of you don;t have children, you get a Handmaid. I don't know if you'd get a Handmaid if you have kids already though.
Otherwise, if you're on your first marriage to just a normal guy, you're an econowife.
Marthas - I reckon they're older or infertile single women (widowed/divorced/never married) who haven't caused trouble for the regime. The Aunts are recruited from this pool too. If you're older/infertile and you've protested against the regime, or you're a "gender traitor", or a CEO, or a professor of biology, or a doctor - you're either killed outright or sent to the Colonies to clean up radioactive waste. If you're a young and fertile trouble maker, you get to be a Handmaid.