What makes an agency independent
Some agencies answer to the President like any other; others Congress deliberately walled off. The difference comes down mostly to one thing, who can remove the leadership, and when.
Removal is the real dividing line
The President can dismiss a Cabinet secretary or the EPA Administrator at any time, for any reason. An 'independent' agency's leaders can be removed only 'for cause', inefficiency, neglect of duty, or malfeasance, which lets them resist day-to-day political direction. That single difference, at-will versus for-cause removal, is what 'independence' usually means.
Multi-member, staggered, often party-balanced
Many independents are commissions of three, five, or seven members serving fixed terms that don't all expire together, frequently capped so that no more than a bare majority comes from one political party. No single President appoints the whole body at once, which keeps any one administration from capturing it overnight.
A few are insulated by their funding, too
A handful fund themselves outside the annual appropriations Congress controls, the Federal Reserve from the interest it earns, the CFPB from Federal Reserve transfers, the FDIC from bank premiums, the Postal Service from postage. Because money is leverage, self-funding removes one more lever of political control.
'Independent' isn't all-or-nothing
The EPA sits outside any department yet its Administrator serves at will. The CFPB funds itself yet its single Director became removable at will after Seila Law. Independence is a spectrum, set by removal protection, structure, and funding together, not a yes-or-no label.