Welcome to the CMMI! The great thing about this business is that new people are always coming into it! Sometimes we get so wrapped up in debating esoteric points that we forget that new people are wondering what the heck we're talking about!
The different CMMI levels describe the process performance behavior of an organization starting with a chaotic state and progressing upward. Without getting in to all of the details of Maturity Levels and Capability Levels (all things you will learn when you take any "Introduction to CMMI" class), let me try to paraphrase it and net it out for you.
A Level One organization is pretty chaotic and reactive, responding to defects and issues as they arise. They typically experience a high level of defects discovered late in the engineering cycle, miss delivery dates, and underestimate the project effort. Sound familiar?
A Level Two organization demonstrates some basic process management at the project level - specifically around the project management (PP, PMC), measurement (MA), Configuration Management (CM), QA (PPQA), Requirement Management (REQM), and Supplier management (SAM) areas. There will be policies, plans, some training, some monitoring, and some review by management (among other things) of the process, albeit at the project level and perhaps not consistently across the organization.
A Level Three organization has invested in a process infrastructure, has a set of standard processes from which projects can choose, uses an engineering life cycle, continuously improves the process, and manages risk. There are eleven additional process areas in Maturity Level Three (including all of the engineering processes) and the whole thing is managed like an overall process program.
A Level Four organization has a statistical understanding of the way their process performs. They are able to understand what occurs when a particular sub process is employed, and they are able to predict what WILL happen (based on data) if a sub process will be used.
A Level Five company uses that statistical data to continuously improve the process and to bring predictability to the introduction of innovations and improvements to the organization.
A good Introduction to CMMI instructor should be able to elaborate on all of this. Good luck!