Consider the risk
Another way of approaching this question is by considering the risk involved in a project. Think about a software product like a website and consider the risk to the project and your business if the product were to not function as intended. Risks like that of an issue causing some mild customer frustration are more likely but have a relatively low severity whereas damage to brand reputation or loss of data and/or sales have a significantly higher severity but are generally much less common. Testing cannot guarantee these issues do not exist but it can reduce the chance that any issue exists and hence reduce the project risk. The question is can you afford not to test? Can you live with the risk that exists without testing? For extremely simple projects, where the potential risk is already very low then perhaps you can live with it. But for most projects where a serious failure would have a significant impact on the business, I'm guessing you will want to do what you can to minimise that risk.
Who should be testing?
Perhaps as important as doing testing, you need to also consider who is doing the testing. It is broadly true that anyone can test software, and quite often many of the stakeholders involved in a project will do testing to some extent. The developers will test their code looking for errors, the project manager will evaluate the product to make sure it matches their intent and understanding, and the client will likely also want to evaluate the product before release. A skilled tester will also check these things and provide another set of eyes but more than that a skilled tester is practiced at experimentation and observation. I'm sure most of us will remember performing experiments in science class at school but this isn't really comparable to the work that people in white coats who work in a lab do. There are few who think that the later is anything less than incredibly complex, methodical, and can only be carried out by people with a specific skill set. In the same way most, if not all, are capable of testing software in some sense. Anyone can click around a website, try a few things and if it's a new product you will probably find some bugs, but this only just begins to scratch the surface of software testing.
Skilled software testers devote a lot of time developing the skills and knowledge that will help them become better at this. However, much of the knowledge required to be a skilled tester is the kind of knowledge that is difficult to transfer to another person by means of writing it down or verbalizing it. Much like the ability to speak a language or perform improvisation, much of testing cannot be learned from reading about testing and this tacit knowledge has to be developed through practice. Skilled testers will be able to think of questions and test ideas that no one else on the team has, and in addition will often observe behaviors others will miss.
