Mint used to use a company called Yodlee to do data aggregation. You can find more information here: http://www.yodlee.com/developers-and-alliances/yodlee-apis/

Since Mint's acquisition by Intuit, they have switched over to Intuit's in-house data aggregation system. Intuit is planning to release these APIs to developers some time in the next few months. You can find more information here: http://developer.intuit.com/agg-cat/

There are some other companies out there like CashEdge and ByAllAccounts, but they tend not to do deals with startup companies and instead focus on selling to big financial institutions like banks, etc.

You also might want to check out Wesabe's (former Mint competitor) data aggregation system which they have open-sourced: https://github.com/wesabe/ssu