a. The applicants for the job either had a first class degree or some teaching experience.
b. All competitors must either be male or wear a one-piece swimming costume.
In each of these cases, the implication that or contributes to the sentence as a whole is that one of the conjuncts is true. In (1), the applicants are implied to have had either a first-class degree but no teaching experience, or teaching experience but not a first-class degree, or possibly both. That is to say there is an interpretation in which both implications can be held simultaneously. In (2), the implication is similar. The utterer of such a sentence would certainly not be excluding the possibility of both of the conjuncts being true, for this would imply that a male competitor had either to wear nothing a two piece costume! On the contrary, the sentence allows the following types of competitor: male competitors (whether wearing one-piece costume or not) and non-male competitors wearing one piece costume. This disjunction in the characterization of or can be stated more formally in terms of truth conditions.
P and Q each represent sentence and v correspond to or. P v Q will be true if and only if either P is true, or Q is true, or P and Q are true. It can be true under different conditions.