A counterparty is a party with which a transaction is done. If A sells something to B, then B is a counter-party from A's point of view and vice-versa.
The risk that the counterparty will fail to fulfil their obligations - usually either by failing to pay or by failing to deliver securities - is called counterparty risk.
Financial institutions should track and manage counterpart risk in much the same way as any other credit risk, and this should be integrated into institutions' overall risk management system.
The counterparty risks from securities trading are either simple credit risks (where the risk is that the other party will not pay) or a combination of credit risk with the risk of a position in a derivative (where the risk is that the other part will not deliver securities).
Counterparty risk tends to be at least as much of a concern to regulators as to the institutions exposed to it. This is because a large financial institution will be a counterparty to many others, and therefore the knock-on effects of its failure pose a systemic risk.