I know that the general way of saying ‘let him/her/them’ is «пусть он/она/они» and the general way of saying ‘let us’ is «давай(те)». (‘General’ meaning that I'm aware that sometimes when translated, the verb is used in the appropriate form and the word ‘let’ is omitted; I am referring to instances apart from these.) What is the general way of saying ‘let me/you’? I'm not sure if it involves some form of «позволить» and how generally «позволить» can be used to say ‘let [someone]’. Or would it be better to rephrase the sentence to avoid using the word ‘let’?
Well, anyway, let us begin at the beginning. The word "let" may mean the following (and totally different) things:
- A proposal. Translated as "давай(те)" + future tense.
Example: "Let us go somewhere" - "давай(те) сходим куда-нибудь".
Note: "давай(те)" could be omitted, so the informal things become more informal, while the formal things become even more formal. For example, "Let us suppose" - "Давайте предположим" (a mixture of common and formal speech, typical "lecturer" tone), or just "Предположим" (neutrally formal, like in a book).
- An indifference (let it be / I don't mind / I don't care). Translated as "Пусть" or "Пускай" + present tense.
Example: "Let things go" - "Пускай всё идёт своим чередом".
- A rhetorical request. Typically translated as "Позвольте" or "Разрешите" + infinitive.
Example: "Let me recall that..." - "Позвольте напомнить, что..."
Note: obviously, it belongs to a purely formal speech.
- An order or persistent request. Not translated, i.e. imperative only.
Example: "Let him go" - "Отпусти(те) его". (BTW. If you really mean "let him go, because I don't care", you should say "Пускай уходит", see (2)).
Note: as usual, an order differs from a request by tone, and by using the word "please" and such.
- A polite request (May I...). Well, now we're entering the jungle of a speech etiquette, so we have first to ask if "let" is really a polite request in English. How big is the difference between "Let me help you" and "May I help you?", or "May I come through?" and "Let me come through"?
IMO, the most common and simple way in Russian to form a polite request is to use "Могу" (or "Можно") + infinitive, or, alternatively, "Можно" + future tense (also note that it's a question, so it directly corresponds to "May I?" type of sentences). However, the more direct approach with "Дай(те)" + infinitive (which is much more like "Let me...") is generally also acceptable (at least if you don't forget to add "please"). So we have
Example: "May I come through?" - "Можно пройти?"; "Let me come through" - "Дайте пройти".
Considering usage of "Позвольте" in a live speech, it's definitely an option if you want to sound more like a gentleman, and many people actually say it, but it's not a first choice by any means.
Also, there are specific use cases, where the construction "Let + verb" is used to denote a single action, which usually correspond to a single verb in Russian, e.g. "Let know" - "Сообщить" (although here it's also possible: "Дать знать"), "Let fall" - "Уронить", and so on.
Tough question because the English "let" is an intersection of several constructions in Russian, and one probably can't avoid asking "which 'let'".
Anyway, it's not позволить. It can be, but only in a handful of hyper-formal contexts, and in general you're better off thinking of позволить as to "allow" or "enable" rather than "let".
In everyday speech, you can safely use давай or plural/formal давайте with "me" and "you"; just include the pronoun. For a less abstract "how about this is what happens next" and more of a literal "let", you can use дай(те), which is even more informal but also has an undertone of asking permission.
To expand a little on Nikolay's answer
let me is
Let me see -> Дай(те)(-ка) (мне) посмотреть / Дай(те)(-ка) я посмотрю *
Let me pass -> Дай(те)(-ка) (мне) пройти / Дай(те)(-ка) я пройду *
Let me think -> Дай(те)(-ка) (мне) подумать / Дай(те)(-ка) я подумаю *
* elements in parentheses are provisional
However it may sound impolite if it implies some action to be taken by the addressee and not just giving their consent, and if the situation doesn't warrant impoliteness or overfamiliarity
разреши(те) can be used instead. The particle
-ка softens the phrase so can be added to the verb to make it sound less categorical and more amiable.
Some fixed expressions with
let have a variant of (or the only correct) translation using a specific verb in imperative mood.
Let me in -
(В)Пусти(те) (меня)along with
Дай(те) (мне) за-/войти
Let me go - (От)Пусти(те) (меня)
Don't let me down - Не подведи(те) (меня)
The verb is not fully translatable into Russian, because in English it primarily conveys both the meaning of a suggestion (imperative, including 2nd and 3rd persons) as well as an ask for a permission, while in Russian the permissive and imperative roles are different, plus there are subtler connotations of lest. The main possible structures as ranged from higher to lower speech registers would be:
- Позволь(те) мне (нам, ему, ей, им) + infinitive (also in Negative Imperative), or Позволь(те), я (мы, он(и/-а) + analytical future tense for negative imperative and synthetical future tense for affirmative ones:
Позвольте нам не вовевать. Позвольте нам успокоиться.
Позвольте, мы не будем воевать. Позвольте, мы успокоимся.
The verb can also be used for contradictions (especially with но), also in Present:
Позвольте, но ведь вы-то воюете!
The pronouns can be usually dropped off with infinitives, but not with personal forms.
The verb is never used in the 2nd person.
- Разреши(те) мне (нам, ему, ей, им) + infinitive (also in Negative Imperative), or Разреши(те), я (мы, он(и/-а)+ a synthetic future (positive statement) or analytical future (negative statement);
Разрешите нам успокоиться. Разрешите нам (не) воевать.
Разрешите, мы успокоимся. Разрешите, мы (не) будем воевать.
The forms are never used with 2ns person, either. This verb cannot be used in contradictory connotations. The pronouns can also be omitted with infinitives, but not with personal forms.
- Дай(те) мне (нам, ему, ей, им) + infinitive or synthetical future form (never used in negative forms);
Дай(те) (нам, ему, ей, им) успокоиться. Дай(те), успокоимся.
- The simpler form, goes as imperative with any person and tense, for negative and positive statements alike;
Давай(те), мы (мы, он(и/-а) + (durative) infinitive or any form of a future tense;
Давай успокаиваться. Давай(,) не (будем) воевать. Давай, успокоимся. Давай(те)(,) вы успокоитесь. Давай(те)(,) вы не будете / ты не будешь воевать.
- Пусть + any form of a future tense describes any form of indirect (that is, non-evidential) imperative:
Пусть мы (не) успокоимся. Пусть мы (не) будем воевать. Пусть ты (не) успокоишься. Пусть ты (не) будешь воевать. Пусть они (не) успокоятся. Пусть они (не) будут воевать.
- The modal structures of let-me-go!-type are usually rendered by occasional verbs or structures:
Let me go! = (От)пусти! / Отвянь! Отдзынь!, etc.
Let's go! = Начали! Начинаем! Поехали! Понеслась! Вперёд! Идём!, etc.
Let's get it! = Давай-давай! Так (и надо)! Эшкере ! (the latter is used as a loan interjection in teenage speech)