Testing should be an important part of any web project, to ensure a quality delivery.
There are many types if testing. The most common is acceptance testing, which is a test to see if the delivery is according to the contract/specification. How rigorous the acceptance test is, varies a lot from project to project, and you can in many projects influence how much effort is spent here. In our experience, you save a lot of time and money if you discover bugs and problems in this phase rather than in production. So we recommend a fairly diligent acceptance test.
Compatibility testing is sometimes a part of the acceptance testing, but if not, this should be tested anyway. Does the website work in the different browsers agreed upon? Is it compliant with accessibility standards?
Performance testing is also often a part of the acceptance testing. This often involves testing page load times and scalability.
Usability testing is testing where you put real users in front of the system, and analyze what they do, and see if they have any problems navigating or using the system.
We also suggest you go through our Pre-go-live checklist as part of the testing process.