The email services I have come across with robust API integration are:
I'm using CritSend for my web app (approx. 25000 users) and it works well. SendGrid and CritSend are priced very similarly, though CritSend is basically a buy credits then use them model, whereas SendGrid requires you to pay a basic monthly fee and then pay extra if you exceed your quota. It works out basically the same, but with SendGrid you have to pay a minimum amount each month (which still counts towards email credits) whereas CritSend has no minimum. Both services offer free credits on signing up then charge around $1 per 1000 emails sent.
I haven't used PostMark, though their pricing is 50% more expensive than SendGrid and CritSend. Their presentation is much more professional/refined than CritSend or SendGrid.
SocketLabs I only heard of because of Brian's answer to this question.
MailChimp have a robust API but are more focussed on campaign sending than individual emails. Their presentation, reporting and setup are all very professional and user friendly.