A "representation on authority of parties/signatories" clause of a contract states that the parties who sign the agreement have the authority to bind the parties to the agreement. When you sign your contract with another party, you aren't asking for this person's autograph — you want the signature to certify that that party has the authority to sign the contract and have it be legally binding.
Signatures may seem to be the easiest part of any contract, but they are botched often enough to require some attention. If you have entered into a contract with a corporation, it is not enough to only obtain the signature of some individual representative. With only that person's signature, it is possible that only that individual and not the corporation itself can be held liable under the contract. It is also possible that you could obtain the signature of a person who doesn't have the authority to sign the contract. This clause is intended to avoid these possibilities.
Make sure that you obtain the signature of the appropriate individual representing a company. For example, the signature line for a corporation should typically read:
a California corporation
By: Joe Woohoo
Title: Chief Executive Officer
Here is a sample clause:
Representation on Authority of Parties/Signatories. Each person signing this Agreement represents and warrants that he or she is duly authorized and has legal capacity to execute and deliver this Agreement. Each party represents and warrants to the other that the execution and delivery of the Agreement and the performance of such party's obligations hereunder have been duly authorized and that the Agreement is a valid and legal agreement binding on such party and enforceable in accordance with its terms.