A conditional fee (sometimes called a contingent fee) arrangement is a fee for legal services in pursuing a claim which is only payable if and when the claim is successful and costs recovered.
It is one of a number of ways in which an impecunious claimant, who cannot otherwise afford to pursue their case, can have access to justice. The lawyer generally acts on the basis that the claim is a good one, so that there is a good chance of recovery; and the reward will often be specified as a given percentage or proportion of the damages or financial compensation recovered. If the case is lost, the lawyer gets nothing.
Not to be confused with litigation funding, which is where a third party financially supports a claim on the basis that they will be paid a proportion of any financial compensation recovered.