The grammatical terms first/second/third person are opaque (in the sense that, if you don't already know the meaning, you'd be very lucky to guess it). As it happens the terms used in Arabic are much more illuminating:
- First person = "the speaker": the person (or people) talking (or writing), or the group on whose behalf they are talking.
- Second person = "the addressed": the person to whom the speech/writing is directed.
- Third person = "the absent": someone who is neither speaking nor being spoken to.
These terms are used to classify words according to who they refer to: as you mention in the question, when it comes to pronouns, I and we belong to the first person, you to the second, and he/she/it/they to the third person. But there are other uses for the terms too - for instance the verb form goes is exclusively used for the third person.
(That's more than enough about the use of the terms in grammar.)
The same terms have been borrowed from grammar to describe certain types of narrative in literature:
- A first person narrative is one told from the perspective of someone who is inside the story (Reader, I married him). This name derives from the fact that such a narrative will often use forms such as "I saw ..." and "It made me feel ...", but note that by no means all of the verbs and pronouns in such a narrative will be first person - there will usually be plenty of third person forms too. Such a narrative will only give the perspective of that one character - so, for instance, it will not describe the thoughts and feelings of other characters, unless they are apparent to the narrator. (This is true in general, but of course some authors do like to play with the form and include observations that the narrator character would not be privy to.)
- A third person narrative, by contrast, is told from the perspective of someone outside the story (There was a boy called Eustace Clarence Scrubb, and he almost deserved it) - usually someone omniscient, who can see into the thoughts, feelings and motivations of all of the characters. Usually all of the pronouns and verbs (other than those in quoted speech) will be in the third person.
The term second person is not commonly used in this sense. (It might make sense to do so for some of those "adventure game" books, where you get to choose your own ending - You are in a long corridor. Turn to page 25 to go east, or 132 to go west - but this is not a standard usage.)
Moving on to video games:
- First person, by analogy with the first person narrative, refers to a game in which the perspective given to the player is that of one of the characters inside the game's story.
- Third person, then, refers to a view that is removed from any of the characters that exist within the game's story - a bird's eye view of the game's world, or a perspective view of it from a camera - would qualify for this definition.
Once again, second person is not commonly used: it could perhaps be apt for text adventures (You are in a maze of twisty passages, all alike... You are likely to be eaten by a grue), but this is certainly nonstandard.