~kudasai is no doubt a word that you have heard before if you have ever partaken of any Japanese media, where it is typically translated in dubs or subtitles as "please (would you this favour for me)", but Japanese has more expressions and words like these and we will be going through most of them in this post.

This category of verbs are known as benefactives, that is words of a modality that conveys a wish or in cruder terms a politely phrased imperative. Hence, ~nasai as in gomennasai (御免なさい) is literally "Do please forgive me," with nasai being the formal imperative form of nasaru (なさる, "the sir/madam does"), a reverential (sonkeigo) verb, but more than often the meaning of gomennasai is closer to "excuse me" or "pardon."

This group of words could very well also be considered as "polite imperatives."

We will not, however, be going through humble (kenjougo) verbs like itadaku (頂く, "to humbly receive") in this article since its usage is wider and a bit more different than these benefactives.

-kudasai (下さい) - Derived from kudasaru (下さる, "to deign to give")
-kudasaimase (下さいませ) - More formal version of the above.
-nasai (なさい) - From the nasaru (なさる, "the sir/madam does").
-nasaimase (なさいませ) - More formal version of the above.
-nasaimashi (なさいまし) - Archaic variant of the above.
-asobase (遊ばせ) - Archaic and hyper-formal, derived from asobu (遊ぶ, "to frolic, to play"). More often than not used by courtiers towards their regent when asking either for favours or imploring.
-choudai (頂戴) - Semi-formal, literally a noun meaning "reception", it is literally comprised of two ways of spelling the humble verb itadaku ("to humbly ask for"), the humble, kenjougo, form of morau ("to get").  
-goran (御覧) - Semi-formal, literally the reverential, sonkeigo, form of miru ("to see"), i.e. "the sir/madam sees."
-tamae (給え) - In modern times a masculine and informal impeartive, but I briefly mentioned in a previous post its usage as a archaic benefactive verb in the sense of "superior/god/ruler grants," hence you will still see it in media such as fantasy novels where ~tamae usually rounds off prayer-like commands or incantations.
-(i)rasshai (いらっしゃい) - From the reverential, sonkeigo, existential verb irassharu (いらっしゃる, "the sir/madam is"). A closely related cousin of this expression is irasshaimase (いらっしゃいませ), the more formal if old fashioned version of irasshai, that you will typically hear from shop-owners trying to persuade potential buyers into their shops.

These do in general go at the end of a verb (or a noun, in some cases) to transform it into the imperative or benefactive mood, but in the case of the formal imperatives, the honorific prefix o- or go- is placed to render it more polite, such as:

読める -> お読めください
Yomeru -> Oyomekudasai
"Read" -> "Please read"

Or in the case of noun-based verbs:
了承する ->  ご了承ください
Ryoushou suru -> Goryoushou kudasai
"To understand" -> "Please understand"

In some cases it could also be phrased goryoushu shite kudasai, where the shite is less formal, than the above, but still polite, where the shite is the so-called "te" form, or a combinatory form of sorts to connect verbs and create things such as compounds.