Positive Testing is any activity aimed at evaluating an attribute or a capability of a system and determining that it meets its requirements. Expectation is that valid data is input and will be processed through the normal paths.
Negative Testing is the process of executing a program or system with the intent of finding errors. Expectation is that invalid data is input and will be processed through the error handling paths. This can also include error testing for integration points being unavailable. For example, if your application interfaces with 2 other applications, “disconnect” the connection to one of them to test the error handling path for that integration point being unavailable.