The Associate Software Engineer position is usually reserved for cases where we want to groom people into the Software Engineering career. Typically, these people might come from bootcamps where they know how to code but they can't work independently and require guidance to build a system.
An Engineering Lead is usually the final position before an engineer decides whether to invest 100% into the Manager Track or the Individual Contributor Track. It is common to find Engineering Leads owning all the delivery within a team, managing the engineers in their team and also act as their Software Architect.
A Staff Engineer role is the beginning of an engineering leadership role where you lead without direct management authority. Staff Software Engineers are commonly expected to act as an enabler for at least two teams and think about bigger problems across a team boundary.
These Senior Staff Engineers typically own a much bigger scope than the level below. In some companies, this could be an entire business unit consisting of multiple squads. They form best practices within their own business unit and are expected to collaborate with other business units.
Principal Software Engineers usually look after the architecture of multiple business units (or the whole company) in their portfolio, influencing engineers with strong persuasion and technical ability instead of pure authority. These engineers are usually trusted thought partners of senior leadership like CTOs.