Independent agency vs. executive agency
'Executive agency' and 'independent agency' sound like opposites. In practice the categories overlap, and the labels mislead about as often as they inform.
Executive (department) agencies
These sit inside, or are, a Cabinet department. Their heads serve at the President's pleasure, putting them squarely under presidential control. Most of the federal workforce is here.
Independent agencies, organizationally
These stand outside the departments and report to no Cabinet secretary. But that is an org-chart fact, not a guarantee of insulation: being 'independent' on paper doesn't always mean the head is protected from removal.
Independent regulatory commissions
The strong form of independence: multi-member, for-cause-protected, fixed-term bodies that write rules and decide cases, the SEC, FCC, FTC, NLRB, FERC. These are the agencies the removal-power cases are really about.
Why the labels mislead
The EPA is an 'independent agency' whose Administrator the President can fire at will. A single-director bureau can be 'independent' and self-funded yet still removable at will. The distinction that actually matters is removal protection, not which box the entity occupies on the chart.