In a sense, when you make a promise to someone, you make a covenant. However, a covenant is much more than a sign because it signifies a full commitment to something or mostly to someone. However, usually a covenant is not made in writing, but instead, it is often made verbally like in a wedding ceremony. A contract, on the other hand, is mostly made in writing.
The parties come together to discuss usually a business agreement where one person is the worker, and the other person needs work done. Then they sign the contract. A covenant, on the other hand, is not signed. A contract can be verbal, but most people try to put the contract in writing so that everyone knows the terms of the contract.