What is Test Case?
A test case is a document that describes an input, action, or event, and its expected results, in order to determine if a feature of an application is working correctly.

A test case (how to test) is a document which consists of a set of conditions or actions which are performed on the software application in order to verify the expected functionality of the feature.

Some elements of Test Case:

Test Suite ID: The ID of the test suite to which this test case belongs.
Test Case ID: The ID of the test case.
Test Scenario: A Test Scenario is defined as any functionality that can be tested. It is a collective set of test cases .
Test Procedure(Test Steps): Step-by-step procedure to execute the test.
Test Data: The test data, or links to the test data, that are to be used while conducting the test.
Expected Result: The expected result of the test.
Actual Result: The actual result of the test; to be filled after executing the test.
Status(Pass/Fail): After comparing with both expected result & actual result the tester( creator of test case) evaluate weather the test case is Pass/Fail.

What is Bug Report?
A Bug Report in Software Testing is a detailed document about bugs found in the software application. Bug report contains each detail about bugs like description, date when bug was found, name of tester who found it, name of developer who fixed it, etc. Bug report helps to identify similar bugs in future so it can be avoided.

The key elements of a bug report in Software Testing:

Title: An appropriate title is simple, brief, and provides a description of what the bug is to the developer.
Description: This is a description, written in brief, of the bug and how and why it happened. More specifics than the title should be included in this part.
Reproducing steps: This must include the maximum steps required for the bug to repeat. The steps should ideally be brief, quick, and everyone can follow them.
Environment: This section should provide all the specifics of the setup and configuration of the environment that the app is running on.
Expected Result: This is the result of an activity that was anticipated or expected.
Actual Result: This is the result observed by the tester or user.
Attachments: Attachments can be very useful for the developer to recognize the problem faster.
Severity and Priority: Severity and Priority is a measure of how important and how serious an issue is.
Assignee: To whom the bug is allocated.