Executive search firms work for the company that hires them to fill their vacant position. That company, and not the individual looking for work, is their client. The search firm's goal should always be to fill the position successfully for their client, and not to find work for one of their candidates.
The search firm must understand the position in order to find the right candidates. Most frequently, a representative from the search firm works together with someone from the searching company to develop a written Job Specification. The search firm then begins the process of finding the right executive to meet those specifications.
There are two types of executive search firms: contingent and retained search. A contingency firm only earns a fee for its service if their client hires a candidate that the contingency firm presents. There is no exclusivity with a client's relationship with a contingency firm -- the client can work with multiple executive search firms at once, and/or can continue to seek out candidates on its own.
The term "headhunting" is most often associated with contingency firms, rather than retained search firms. With contingent search, the emphasis is generally less on finding the exact right executive for the position, and more on finding as many qualified potential candidates as possible. The recruiter wants to present the client with a large number of resumes, about which the client company can make its own decision.
The other types of executive search, retained executive search, is typically an exclusive relationship between the client and the search firm. The two sign a contract which defines their relationship and each party's expectations. The search firm is paid for its time, rather than being paid only if the client chooses one of its candidates.
With a retained executive search firm, the recruiter has a higher level of responsibility for obtaining the right "fit" between the executive hired and the company hiring. The search process is highly structured, and the retained search firm in general bears more of the responsibility for the process than with a contingency firm.