WHAT?
There are lot's of definitions for CMM. One good one is this:
It defines how software organizations mature or improve in their ability to develop software.
The model was developed by Software Engineering Institute (SEI) of CarnegieMellonUniversity in late 80s. This model provides a structured view in a five-layer model of increasingly sophisticated practices for those working in software. Each level (except Level 1) has certain Key Process Areas associated with it. Each level addresses levels of maturity exhibited by the project. The first version of CMM came in early 90s.
About 3 years back, SEI took up an integration project. There were multiple CMM models in place - one for software development, one for Integrated process and product development, one for systems engineering etc. Organizations (and SEI!) found it difficult to live with multiple models. Hence this project integrated the different CMM models into one and thus CMM - Integrated , CMMI came into being. The structure of CMMI is similar to CMM. The plan is that CMM will be removed within a year or so.
WHY?
CMMI describes how software organizations can take the path of continuous improvement which is so required in this highly competitive world. Moreover, it is a model that can be tailored to suit the organization. "Keep Improving" - is the CMMI mantra. CMMI also creates brand value and many clients are demanding it.