An agreement is the consent of two or more persons to form some engagement, or to rescind or modify and engagement already made. Dusrum vel plurium in idem placitum consensus. That kind of agreement, the object of which is the formation of an engagement, is called a contract.
Per Mance LJ in Baird Textile Holdings Ltd v Marks & Spencer Plc  EWCA Civ 274 at paras 59-61:
59. … For a contract to come into existence, there must be both (a) an agreement on essentials with sufficient certainty to be enforceable and (b) an intention to create legal relations.
60. Both requirements are normally judged objectively. Absence of the former may involve or be explained by the latter. But this is not always so. A sufficiently certain agreement may be reached, but there may be either expressly (i.e. by express agreement) or impliedly (e.g. in some family situations) no intention to create legal relations.
61. An intention to create legal relations is normally presumed in the case of an express or apparent agreement satisfying the first requirement: see Chitty on Contracts (28th Ed.) Vol. 1 para.2-146. It is otherwise, when the case is that an implied contract falls to be inferred from parties’ conduct: Chitty, para.2-147. It is then for the party asserting such a contract to show the necessity for implying it. As Morison J said in his paragraph 12(1), if the parties would or might have acted as they did without any such contract, there is no necessity to imply any contract. It is merely putting the same point another way to say that no intention to make any such contract will then be inferred.