Software engineering management can be defined as the application of management activities — planning, coordinating, measuring, monitoring, controlling, and reporting — to ensure that software products and software engineering services are delivered efficiently, effectively, and to the benefit of stakeholders.
— http://swebokwiki.org/Chapter_7:_Software_Engineering_Management
Source: SWEBOK
Software engineering is a field within computer science focused on designing, developing, testing, and maintaining of software applications. It involves applying engineering principles and computer programming expertise to develop software systems that meet user needs.
The terms programmer and coder overlap software engineer, but they imply only the construction aspect of typical software engineer workload.
A software engineer applies a software development process, which involves defining, implementing, testing, managing, and maintaining software systems and, creating and modifying the development process.