As is often the case, there are competing definitions as to what constitutes each level. If there is a particular model you are attempting to follow you will find the definition(s) vary slightly but follow a general theme. I have tried to outline below:
Level One: Initial - Testing is Ad-Hoc. In particular during a crisis, procedures are ignored or not implemented. Often under the guise of "to be faster"
Level Two: Repeatable - Testing is basic, but planned. There might be test strategies, test plans, test cases which are run at the end of a project based on requirements. Normally, a waterfall model is followed at this level.
Level Three: Defined - Testing follows a SDLC with specialist testers carrying out tests throughout the cycle. A simple example is the V-Model where testing is involved at a number of touchpoints. This is also the first level to take into account Non-Functional testing.
Level Four: Managed - Testing is carried out at every stage of the SDLC, including reviews of requirements and designs. This is often considerd the first level of true agile testing. Testing is peer reviewed and external stakeholders are involved at each stage.
Level Five: Optimizing - The ideal. This is where the testing process itself is tested and improved. It is often said that at this level "everyone is a tester", with everyone on the team contributing to the testing and process improvement.