Contingent/Contingency Fee Agreement
When an attorney agrees to represent a client in a civil lawsuit in exchange for a percentage of the settlement without requiring payment of a retainer or fees up front, s/he is doing so under a contingent fee agreement.
This can be an ideal arrangement for those who are seeking a legal remedy for damages and cannot afford the fees charged by attorneys (often anywhere from $180 to $600 per hour). In addition, it gives one's legal counsel motivation to represent the client to his/her best ability, since the attorney collects payment for services rendered (as well as related expenses, such as hiring expert witnesses) only if s/he wins the case.
The usual percentage in a contingency fee agreement is 25 - 40%, depending on several factors. If the case is extremely complicated and there are sizable punitive damages awarded, this percentage can go up to as much as 50%.
Working on a contingency basis also motivates an attorney to settle out of court, since time is money. If a resolution is negotiated before the defendant responds to a complaint, the attorney's fee may be as low as 25%. If the complaint has been served but expert witnesses have not yet been hired by the time an agreement is reached, the usual fee is around 33%. Once the services of expert witnesses have been retained and they have made disclosures (given testimony under oath) in pre-trial depositions, the percentage rises to 40%.
Every reputable law firm as a written contingency agreement that the client will be expected to read and sign; you can view many of these online by typing the phrase "contingent fee agreement" into any search engine.