'Or' is a conjunction that indicates that at least two things are possible: "apples or oranges" indicates that you have a choice of having an apple, and a choice of having an orange. It is a common logical operator in logic systems.
In the weak (or inclusive) sense, 'or' is used to indicate that at least one, and perhaps more, of the given options could be. ("Do you want an apple or an orange?" could be answered "an apple", "an orange", or "both!") For clarification, sometimes 'and/or' is used.
In the strong sense (AKA the exclusive or, xor), 'or' means that there are two possibilities, only one of which can be ("It will be 10 or 15 minutes.").
In symbolic logic, 'or' is used in the weak sense, and is called a disjunction. It is symbolized by a lowercase 'v' (or ∨ if your browser supports it) called a wedge or a vee. This is an abbreviation of the Latin word for the weak sense of 'or', vel. (Latin also had a word for the strong sense, aut).
If you want to use the strong sense of 'or' in symbolic logic, you have to go the long way around;
(p or q) and not (p and q).
(p ∨ q) ∧ ~(p ∧ q)
Where ~ stands for 'negation' (standard symbol) and ∧ stands for 'and' (in symbolic logic, a bullet point is often used, but this upside-down arrow is also common).