Process Maturity - in an organisation means, all the processes and/or procedures are done in a documented way and everyone knows what is expected of them and performs accordingly rather than ad hoc or random activities. It gives an indication of how close a developing process is to completion and capable for continual improvement.
CMMI (Capability Maturity Model Integration) developed by Carnegie Mellon University is a very common model used to assess the maturity level of a process / department / organization.
An application oriented question on the topic along with responses can be seen below. The best answer was provided by Rajesh Chakrabarty on 7th November 2017.