An allocution, from the Latin for 'to' + 'speak', was originally an address from a Roman general to his soldiers; from which it was transferred to that of a Pope to his cardinals. The adjective allocutive seems to be a recent coinage in English, as it's not in my dictionaries. A Web search finds examples in relation to addressing an audience in the fields of advertising or broadcasting, but the linguistically interesting use of the word is to describe a strange phenomenon in Basque grammar.

Basque has no gender at all, with the exception of allocutives. There are no noun classes based on sex, and there are no separate words for 'she' or 'he'.

Basque has an intimate second person pronoun hi, the equivalent of tu in Romance languages like French and Spanish. But its use is much more limited. It's really only used between intimate friends of the same sex and age group, and between siblings. Optionally to small children. Unlike in Basque's neighbours French or Spanish, it's not used for spouses, God, or animals. So it has pretty restricted use, but it has the unusual property of making allocutive distinctions of gender.

And what is an allocutive distinction of gender? Well, other languages have different feminine and masculine words for 'you'. Arabic for example: 'anta if you're male and 'anti if you're female. But they're only used if you're in the sentence: e.g. You like ice-cream, I like you, Is this your book?, She gave you an apple. In a language like Arabic the sex of the person mentioned is grammatically marked.

In Basque the sex of the person addressed is. So if I say to you: I slept well last night; He rang her up; They play the flute badly; -- then I'm not mentioning you. But if I'm on intimate hi terms with you I'm obliged to add extra endings that mark whether you are female or male. This only occurs in main clauses, but is obligatory there.

The simplest case is where you are mentioned in the sentence, as the subject of a transitive verb.

    ikusi dut        I saw it
    ikusi duk        you saw it  (male intimate)
    ikusi dun        you saw it  (female intimate)
    ikusi duzu       you saw it  (normal)
    ikusi du         she/he saw it
The intransitive subject doesn't make this gender distinction, though there is a hi form:
    joan naiz        I went
    joan haiz        you went  (intimate, either sex)
    joan zara        you went  (normal)
    joan da          she/he went
It is made when you are the indirect object:
    joan natzaik     I went to you  (male, intimate)
    joan natzain     I went to you  (female, intimate)
    joan natzaizkizu I went to you  (normal)
    joan natzaio     I went to her/him
It is already unusual to have the gender distinction of 'you' in these forms, since Basque has no gender elsewhere, but at least the sentences above contain the bearer of gender. Now consider these:
    joan da          she/he went  (normal)
    joan duk         she/he went  (you're intimate male)
    joan dun         she/he went  (you're intimate female)

    joan naiz        I went  (normal)
    joan nauk        I went  (you're intimate male)
    joan naun        I went  (you're intimate female)

    ikusi dut        I saw it  (normal)
    ikusi diat       I saw it  (you're intimate male)
    ikusi dinat      I saw it  (you're intimate female)
And it goes on like that throughout the conjugation of the auxiliary (naiz ~ du ~ da) and also the handful of other verbs that take finite verb marking.

The allocutive forms show a great variety of regional variation, so the forms I'm quoting are some kind of standard Basque.

Historically the hi is the second person singular, and the ordinary word for 'you', zu, began as the second person plural (as in many other languages: English you, French vous). In modern Basque zu is singular but has general plural agreement, and it has a plural zuek with extra plural markings.

The allocutive forms are old: they go back as far as the earliest known texts. I said above the normal 'you' word, zu, doesn't make gender distinctions. But, extraordinary as it may seem, some varieties of Basque have created new allocutive forms for use with zu as well.

Dependence on the addressee is not unique to Basque, but the inflectional nature of this is very unusual: the only other similar devices I know of are in respectful modes in some East Asian languages, such as Tibetan and Japanese, which are characterized by choosing different words to show respect to your addressee; though of course the pervasive Japanese verb-ending -masu is itself an addressee-respect inflection like that of Basque.

A note on the translation. I have translated all these (which have the present tense of the auxiliary) as simple past, 'I saw', 'I went'. In fact these only apply to events earlier today. A different tense of the auxiliary is needed for events before today. The forms I've used could also be translated 'I have seen', 'I have gone' etc.

Saltarelli, M. (1988) Basque, Croom Helm
Trask, R.L. (1997) The History of Basque, Routledge