A logophor is, roughly, a pronoun used to express the perspective of the person it refers to. The name was originally coined for the distinct pronouns some West African languages have that make these two sentences unambiguous:
(1) Maryi said that shei would come tomorrow.
(2) Maryi said that shej would come tomorrow.
In English it's ambiguous whether 'she' refers back to Mary or to some other female; these two possibilities are indicated here by the choice of same or different indices i, j.

The term 'logophor' was then extended to situations in other languages where the usual rules of binding do not apply. It was observed that almost all of these exceptions involved perspective. See under binding for fuller details of the rules, but in a nutshell the anaphor (or SELF) words have to be bound to their antecedent earlier in the same clause:

(3) Maryi cut herselfi yesterday.
But they cannot be used in a complex sentence when the antecedent is in a clause higher up; in this case the plain pronoun must be used, and this is always ambiguous between same and different person readings. So (4) is ungrammatical and the grammatical equivalent (5a) can also be read as (5b):
(4) *Maryi said that John cut herselfi yesterday.
(5a) Maryi said that John cut heri yesterday.
(5b) Maryi said that John cut herj yesterday.

However exactly the domains for binding are defined, there are uses of SELF pronouns that just don't fit, either because they're a direct violation of the rules, or because there's no antecedent at all to bind to. So (4) in normal intonation is wrong, but by putting focus on the logophor you can read it grammatically:
(4') ?Maryi said that John cut herSELFi yesterday.
I've put a question mark against that one, because while it's not obviously ungrammatical (marked with *), people's judgements will vary on how acceptable it is. This is fairly common with many logophoric uses: there's something other than plain syntax going on, at least in English. It is related to discourse structure as well. Often you can't just read the sentence, you have to set up the right context for it.

Here are some more examples of logophors. What they mostly have in common is some particular focus on a person, or how the sentence is seen from that person's point of view. This is arguable, and the exact nature of logophoricity is still unknown, or even whether it's all a single phenomenon. Is there a common thread to the 'exceptions', or are the rules of binding just not understood yet?

(6) What about myself?
(7) John invited Mary and myself.
(8) There was no-one there apart from myself.
(9) A linguist like myself is often accused of making examples up.
(10) I can't see it myself.
(11) Pictures of myself would look good on this wall.
Conversely, there are situations where the pronoun seems to be in the right position to be a SELF word, following its referent and in the same clause, but unexpectedly we get the SELF-less pronoun instead. What do you feel is the difference in the following pair?
(12a) Johni saw a snake near himi.
(12b) Johni saw a snake near himselfi.
I would hazard a guess that most people feel that the 'him' form is (contrary to the prediction of binding theory) the normal way of saying it; and the 'himself' either sounds odd or has to be focused or contrastive (if he saw a coyote near Bill and a snake near himself), or perhaps that with (12a) you're seeing it through the speaker's eyes and with (12b) you're seeing it through John's eyes. So here again we get an effect of perspective.

The acceptability of co-referential 'him' rather than 'himself' is subject to certain conditions: it needs to appear in an expression where a point of view is physically possible and reasonable. So you can't say it in pairs like the following, which have to have the ordinary anaphoric 'himself':

(13a) *Johni sent a parcel to himi.
(13b) Johni sent a parcel to himselfi.
(14a) ?*Johni spilt a drink on himi.
(14b) Johni spilt a drink on himselfi.

A standard overview of technical work on anaphors and logophors is Reinhart and Reuland, 'Anaphors and logophors: an argument structure perspective', in Koster and Reuland, 1991, Long-Distance Anaphora, Cambridge.

The problem will be solved with lapidary clarity if I ever do my dissertation on it.