Definition A
A contingent fee or contingency is an amount (usually a percentage) of funds paid to an attorney or other professional as part of the money awarded to a business as a result of a verdict. A contingency is only awarded to the attorney if the lawsuit results in a payment to the client or is favorably settled out of court. If the client receives no payment, the attorney does not receive anything.
Contingency fees are helpful in allowing businesses and individuals to bring a suit without having to pay expensive attorney fees. They also encourage the attorney not to take a case which is not winnable and to work diligently on behalf of the client.
In the U.S., contingency fees are most often found in personal injury cases.
Contingency fees are different from retainers, in which the client must pay the attorney a non-refundable fee at the start of the case.
Definition B
A contingency in a business contract is called a "condition precedent." It is an event which must occur before a contract can be finalized. For example, in a contract for the sale/purchase of a business, one contingency might be that the buyer obtains acceptable financing. Another contingency in a business sale might be that there is an acceptable transfer of the lease. If the contingency does not happen, the contract is not executed.

