Public accounting includes any accounting work that a company
performs for another company. Examples would be audits, tax
compliance, consulting, etc. The "Big 4" (KPMG, Deloitte &
Touche, PriceWaterhouseCoopers, and Ernst & Young) are the
dominant firms that provide public accounting services. Private
accounting is accounting work that is done for your own company.
Every company has some form of an internal accounting department
and those employees would be considered private accountants.