Below are the definitions of the various types of recruiters.
A Contingency Recruiting Agency does a employee search on a contingency basis for a company. The recruiter is paid upon a successful hire/placement. The recruiter is responsible to do the initial recruiting, screening and interviewing. The contingency recruiter will also arrange interviews with the candidates for the client/company. The company pays either a flat fee or a percentage of the first years salary. Traditionally, the jobseeker does not pay a fee.
A Retained Agency is similar to a contingency recruiting agency in that the client company pays the fees. Traditionally, the jobseeker/candidates does not pay a fee. The main difference of a retained recruiting agency is that the client company pays a retainer fee to have that company perform a search. A portion of the overall search fee is paid upfront and the remainder is due upon a successful hire. The initial retainer fee is paid whether or not a placement/hire is made. Companies use retained search firms for higher level positions within an organization. For example a CEO, Vice President of Sales, etc.
An Outplacement Agency provides job hunting assistance to jobseekers, typically downsized/displaced individuals. Many times the employer will hire an outplacement company to help their recently downsized workforce find jobs. Outplacement services provides resume/interviewing assistance, career counseling, etc.
Staffing Agency (Temp/Contract)
A Temporary/Contract Staffing Firm hires jobseekers/employees for a client company's temporary/contract hiring needs. In this relationship the jobseeker is employed by the staffing agency and the staffing agency pays all wages, taxes, insurance and benefits. Many times the staffing firm has options for a contract to hire for the employer/jobseeker. The jobseeker traditionally does not pay any fees. The client company pays a premium hourly rate for the contract/temp employee.
Also...several firms will function fully in multiple catagories. RSA (for example) works primarily as a contingency and staffing firm. We do work on retainer occassionally as well and will insist on a retainer for C-level positions.