APIs and OAuth
Firstly, as others have said, you shouldn't be using a user's password to access the API, you should be getting an OAuth token. This will allow you to act on that user's behalf without needing their password. This is a common approach used by many APIs.
Key Exchange
If you need to solve the more general problem of exchanging information over insecure connections, there are several key exchange protocols as mentioned by other answers.
In general key exchange algorithms are secure from eavesdroppers, but because they do not authenticate the identity of the users, they are vulnerable to man-in-the-middle attacks.
From the Wikipedia page on Diffie Hellman:
In the original description, the
Diffie–Hellman exchange by itself does not provide authentication of
the communicating parties and is thus vulnerable to a
man-in-the-middle attack. A person in the middle may establish two
distinct Diffie–Hellman key exchanges, one with Alice and the other
with Bob, effectively masquerading as Alice to Bob, and vice versa,
allowing the attacker to decrypt (and read or store) then re-encrypt
the messages passed between them. A method to authenticate the
communicating parties to each other is generally needed to prevent
this type of attack. Variants of Diffie-Hellman, such as STS, may be
used instead to avoid these types of attacks.
Even STS is insecure in some cases where an attacker is able to insert their own identity (signing key) in place of either the sender or receiver.
Identity and Authentication
This is exactly the problem SSL is designed to solve, by establishing a hierarchy of 'trusted' signing authorities which have in theory verified who owns a domain name, etc, someone connecting to a website can verify that they are indeed communicating with that domain's server, and not with a man-in-the-middle who has placed themselves in between.
You can create a self-signed certificate which will provide the necessary configuration to encrypt the connection, but will not protect you from man in the middle attacks for the same reason that unauthenticated Diffie-Hellman key exchange will not.
You can get free SSL certificates valid for 1 year from https://www.startssl.com/ - I use them for my personal sites. They're not quite as 'trusted' whatever that means, since they only do automatic checks on people who apply for one, but it's free. There are also services which cost very little (£10/year from 123-Reg in the UK).