An Abmahnung (German term for a written warning) is a formal request by one person to another person to forthwith stop a certain behaviour.
If this formal request is made by an attorney, then it is similar to a cease-and-desist letter (but not a cease-and-desist order, which in turn is similar to a einstweilige Verfügung in German law). It is used much more between private parties than a consent decree is in the United States, since German law permits a wide range of parties to file suit to bring an end to an observed wrong done to a third party, whereas in the United States the requirements for a party's standing to sue are more restrictive (i.e. only the wronged party can sue).
In most cases, the law allows the party sending the letter to demand payment of legal costs associated with the sending of the letter. In theory, this allows anybody that observes a violation of a law that may be covered by an Abmahnung to hire an attorney, have a letter sent, and be reimbursed for the attorney's fees. In practice, a lawyer may do this without having been hired by a client who has been wronged, in essence searching for violators of covered laws and demanding payment from them for having found them. Since the enormous proliferation of web sites, this abuse of the law has led to the so-called Abmahnwelle (-wave), as lawyers and copyright holders are trawling the internet searching for even minor and usually unintended violations of e.g. copyright law, with some law firms sending thousands of letter per year demanding payment.