Vayamos vs vamos
I understand that "vamos" and "vámanos" are used meaning "let's go". What I don't understand is why the imperative "vayamos" isn't used.
Why don't we say "vayamos"?
3 Answers
Vamos and vayamos are very similar.
Vamos is more used as a command.
It's late let's go home. Es tarde vamos a casa.
Vayamos is more used as a suggestion.
Let's go to the beach, oh no , please let's go to the beach bar. Vamos a la playa, oh no porfa, vayamos al chiringuito.
And vamonos is diferent. Vamonos means "let's go from here"
I thnk it relates to the second way of forming a Nosotros command:
Vamos + a infinitive, vamos a ir, drop the a ir. So it seems (and I guess is) irregular, but it is logical.
Why is there irregularity, I don’t know- why don’t I say I drinked a glass of water. An etymologist could tell you, but not I.
Take a look at this reference.
http://users.ipfw.edu/jehle/COURSES/CMNDSNOS.HTM
Nosotros Commands
In general:
Nosotros commands are the equivalent of Let's plus a verb in English, for example, Let's dance!. In Spanish, they may be expressed in two ways, using the vamos a + infinitive construction or the subjunctive. Remember that as with all command forms, negatives are based on the subjunctive and object pronouns are attached to the end of affirmative forms.
In the affirmative form the Vamos a + infinitive construction has two meanings. For example, Vamos a bailar can mean We are going to dance (or We will dance), as well as Let's dance, depending on the context or the intent of the speaker. It is safer to use exclamation points to indicate its use as a nosotros command. There are two items to remember, however: 1) the reflexive form is ¡Vámonos! (Let's leave!; note the omission of the first s), and 2) negative forms are based on the subjunctive, vayamos.
¡Vamos a divertirnos! Let's have some fun!
¡ Vámonos! Let's leave/beat it/get out of here!
No vayamos a almorzar ahora. Let's not eat lunch now.
¡No nos vayamos! Let's not leave.
Both explanations given by astotxua and bosquederoble are correct in their statement.
However, as a Spanish teacher, I have been trained in how to explain things to students in a way that is clear concise and plain. At least that's what I try do do. 
So here goes.
The verb "ir" like many other verbs has a reflexive form "irse" The meanings have a slight difference.
"Ir" simply means "to go" while "irse" means to leave, depart "el splitto" (a little humor there)
Normally when a verb appears in "nosotros" command form, the subjunctive is used. i.e., "hablemos" "digamos" "salgamos' and so forth.
The nosotros command form for "ir" is its regular indicative form, or "vamos" This is one of the irregularities of the verb "ir."
As has been stated it can mean "we are going." "we go" or with an infinitive "lets" as in "¡vamos a estudiar!" (Let's study) Also it can simply mean "We are going to study" It depends on the tone of voice.
The negative command "let's not go" will use the subjunctive. "No vayamos."
The reflexive verb "irse" can be used as a statement such as: "nos vamos" --We are leaving/departing, etc.
The command form (affirmative) is "Vámonos" "Let's get outta here!" "Let's leave!"
The negative command will be: "No nos vayamos" "Let's not leave/depart/etc."
I hope that this covers everything. ¡Me voy! I'm outta here!
