A warranty is a statement of a present or historical fact made as a term of a contract. If a warrantor breaches its warranty the injured party might claim damages for the breach of contract and sue for damages, but cannot rescind it altogether. To set aside the contract as if it never happened — to void it, ab initio — you would need to prove a misrepresentation from someone before the contract, that induced you to enter it.
Since a warranty is creature of contract, one’s liability for its failure is absolute: if a warranty fails, you’re in the schtook: it is no defence that you could not reasonably have known that the matters warranted were not true, or that some mendacious interloper (other than the other party to the contract) has intervened to defeat your best intentions unless that kind of conditionality is written into the contract. This is the appeal of a written contract: the parties can write down with infinite, tedious precision, what they mean to say, and what they say they mean, one-hundred per cent.
Representation: A representation is a pre-contractual statement of a present fact, the failure of which entitles an innocent party to rescind a contract altogether (i.e. to put itself in the position it would have been had it never undertaken the ghastly affair in the first place)
Warranty: A warranty is a contractual statement of the truth of a present fact, the failure of falsehood entitles the innocent party to sue for damages for breach of contract (to compensate it for the value of the contract had the statement been correct). It also has a specific meaning as a guarantee of merchantability in the context of a “Manufacturer's Warranty”.
This is why so often you see that most laborious of constructions: “representations and warranties” — the combination giving the wronged party the maximum possible range of remedies.
Promise: This, in Horton Hatches the Egg terms, is what you meant when you said, and what you said when you meant. A promise, if offered, accepted and accompanied by consideration, is your legally binding obligation under a contract.
Undertaking: An undertaking, in a contractual context, is really no different from a promise (though a solicitor’s undertaking is a quite different thing). You might characterise it as a specific promise to ensure that someone else does something (rather than the person making the promise itself), but it isn’t generally used that way.
In any other context, therefore, covenants and undertakings are nothing more than fancy words for promises, and any lover of plain English should therefore seek them out and, with extreme prejudice, relieve them of their command.