FIX = Financial Information eXchange protocol See: http://www.fixprotocol.org/cgi-bin/Welcome.cgi API = Application Programming Interface An API is a (usually documented) set of functions and datatypes that a programmer can use to "link" into another program. For example, IB's TWS has an API that allows a program I have written to send orders through TWS.
FIX was initially created my The Fidelity Institutional Desk in the 90's. It is now a protocol that can be accessed VIA several other vendors. It is used to privately communicate "Indications of Interest", orders and fills privately between the Buy Side trader and the Sales Trader. It was created to replace the bogus indications from AUTEX from the Sell Side and is not to be used unless you actually have an order in that security on your desk. ebo
A FIX connection is just two programs talking to each other using the FIX protocol. One of the programs (the client) might be a black box program that generates buy and sell orders and the other end (the server) could be a clearing firm that accepts and reports fills of those orders. The FIX protocol itself just defines how to "say" things like "I want to buy 5 contracts of XYZ" so that both sides of the conversation understand each other.