What is Use case & Use case testing ?

Use cases:

A sequence of transactions in a dialogue between a user and the system with a tangible result.
Basically, use case is the list of sequential steps that describe how an actor interacts with the system. The steps can be shown in text or as part of a flowchart. And while executing the use case, aim is to find the situation where the system interacts improperly with the user or delivers an improper result.
The use case should also show the results obtained at the end of that sequence of steps. The results obtained should benefit some party, either the actor interacting directly with the system or some other stakeholder who indirectly receives the value of the results.

Use case testing:

A black-box test design technique in which test cases are designed to execute user scenarios.

Conceptually, use case testing is a way to ensure that we have tested typical and exceptional workflows and scenarios for the system, from the point of view of the various actors who directly interact with the system and from the point of view of the various stakeholders who indirectly interact with the system. If we (as test analysts) receive use cases from business analysts or system designers, then these can serve as convenient frameworks for creating test cases.

Assuming we receive a use case, how do we derive tests? Well, at the very least, we should create a test for every workflow, including both the typical and exceptional workflows. If the exceptional workflows were omitted, then you'll need to figure those out, possibly from requirements or some other source. Failing to test exceptions is a common testing mistake when using informal use cases.
Creating tests can involve applying equivalence partitioning and boundary value analysis along the way. In fact, if you find a situation where combinations of conditions determine actions, then you might have found an embedded, implied decision table. Covering the partitions, boundaries, and business rules you discover in the use case might result in 2, 5, 10, 20, or more test cases per workflow, when you're all done.