Software engineering is all about building commercial and mission critical software products cost effectively, within time limits & with good quality. This is achieved by using proven software development processes. These software development processes are also known as methodologies because they try to build a model of the user requirements into a software product (the requirements are the actual reality and its corresponding software product is a model of these requirements).
Some of the popular software engineering methodologies include Scrum, eXtreme Programming, Unified Process etc. Some older methodologies include waterfall model and Rational Unified process.
These methodologies present a standard way of developing a software product. The difference between any of these methodologies is the approach taken. For example in Scrum, the requirements are converted into product features and are kept as a feature backlog. During any iteration, some of the features are selected from this backlog and then they are used to develop the features in that iteration.
If these methodologies are standards themselves then what is the need of process standards? As you might know, some organizations like ISO, IEEE, Software Engineering Institute etc. develop process standards for software development processes.
While any organization may be using any standard methodologies, the organization may not know, how well they are using the methodology. So they may engage a consultant who will assess their processes and conclude how well they are using the processes as compared to industry standards. Their review report may indicate which processes are good and which processes need improvement. Once the organization gets this detailed analysis, it can improve its processes so that they will be at par with industry standards.