Nonprofit Accounting Question:

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 & Touché, Price Waterhouse Coopers, 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.