To understand the Testing Process in an Agile Development Methodology, it is important to understand the Agile Development paradigm. 
 Although the Agile Software Development Manifesto came into existence in February 2001, the concepts existed long before that and were expressed in different ways. Spiral Development Methodology is one such example.

Understanding Agile Software Development:
The Agile Software Development primarily focuses on an iterative method of development and delivery. 
The developers and end users communicate closely and the software is built. A working piece of software is delivered in a short span of time and based on the feedback more features and capabilities are added. 
The focus is on satisfying the customer by delivering working software quickly with minimum features and then improvising on it based on the feedback. The customer is thus closely involved in the Software Design and Development Process. 
The delivery timelines are short and the new code is built on the previous one. 
Despite this, high quality of the product cannot be comprised. 

This creates a different set of challenges for Software Testing.

How is Testing approach different in an Agile Development Scenario?
The Testing Strategy and Approach in Agile Development could be very different from traditional bureaucratic methods. In fact it could vary with project needs and the project team. 
In many scenarios, it may make sense to not have a separate testing team. 
The above statement should be understood carefully. By not having a testing team we do not consider testing to be any less important. In fact testing can done more effectively in short turn around times, by people who know the system and its objectives very well. 

For example in certain teams Business Analysts could be doing a few rounds of testing each time the software version is released. Business Analysts understand the Business Requirements of the Software and test it to verify if it meets the requirements. 

Developers may test the software. They tend to understand the system better and can verify the test results in a better way. Testing for AGILE Software Development requires innovative thinking and the right mix of people should be chosen for doing the testing.

What to test?
Given the relatively short turn around times in this methodology it is important that the team is clear on what needs to be tested. Even though close interaction and innovation is advocated rather than processes, sufficient emphasis is given to the testing effort. 

While each team may have its own group dynamics based on the context, each code has to be unit tested. The developers do the unit testing to ensure that the software unit is functioning correctly. 
Since the development itself is iterative it is possible that the next release of code was built by modifying the previous one. Hence Regression Testing gains significant importance in these situations.

The team tests if the newly added functionality works correctly and that the previously released functionality still works as expected. 

Test Automation also gains importance due to short delivery timelines. Test Automation may prove effective in ensuring that everything that needs to be tested was covered. 
It is not necessary that costly tools be purchased to automate testing. Test Automation can be achieved in a relatively cost effective way by utilizing the various open source tools or by creating in-house scripts. These scripts can run one or more test cases to exercise a unit of code and verify the results or to test several modules. 
This would vary with the complexity of the Project and the experience of the Team

Typical bugs found when doing agile testing?
Although nothing is typical about any Agile Development Project and each project may have its own set of complexities, by the very nature of the paradigm bugs may be introduced in the system when a piece of code is modified/enhanced/changed by one or more Developers.

Whenever a piece of code is changed it is possible that bugs have been introduced to it or previously working code is now broken. New bugs/defects can be introduced at every change or old bugs/defects may be reopened.

Steps Taken to Effectively Test in Agile Development Methodology:
As a wise person once said there is no substitute to hard work. 
The only way one can effectively test is by ensuring Sufficient Test Coverage and Testing Effort Automated or otherwise. 
The challenge could be lack of documentation, but the advantage could be close communication between team members thereby resulting in greater clarity of thought and understanding of the system. 

Each Time Code is changed Regression Testing is done. Test Coverage is ensured by having Automated Scripts and the right mix/combination of People executing the Test Cases. 
Exploratory Testing may also be encouraged. Exploratory Tests are not pre-designed or pre-defined. The Tests are designed and executed immediately. Similarly ad hoc testing may also be encouraged. Ad-hoc testing is done based on the tester’s experience and skills. 

While automated Test Cases will ensure that the Test Cases scripted are executed as defined, the team may not have enough time to design and script all the test cases.

Ensuring software test coverage:
To ensure that delivered product meets the end user’s requirements it is important that sufficient testing is done and all scenarios are tested. 

Sufficient Test Coverage in an Agile Development Scenario may be tricky but with close cooperation and the right team dynamics it may not be impossible. 

The objectives of the Project should be clear to the entire team. Many teams advocate Test Driven Development. At every stage the Software is tested if it meets the Requirements. Every Requirement is translated to a Test Case and Software is validated/ verified. While the Processes and Documentation are not stressed upon sufficient steps are taken to ensure that the software is delivered as per the User Expectations. This implies that each Software delivery should be tested thoroughly before it is released. 

The timelines being short requires that the person testing the software has sufficient knowledge about the system and its objectives