Let's (= let us) + base form of the verb expresses a suggestion.
It's a lovely day. Let's sit outside.
Let's have some coffee (,shall we?).
Let's suggests an action by the speaker and the hearer.
Let's sit outside means that we should sit outside.
The negative is let's
not or don't let's, and for
emphasis we use do let's.
Negative: Let's not waste any time./Don't let's
waste any time.
Emphatic:
Do let's get
started. We've wasted enough time already.
NOTE
The long form is formal and old-fashioned. Let us give thanks to God.
b Let me means that the speaker is telling
him/herself what to do.
Let me think. Where did I put the
letter?
Let me see what's in my diary.
Let me explain.
Let me
think means 'I'm going to think./Give
me time to think.'
NOTE
Let can also have the meaning
'allow'.
Oh, you've got some photos.
Let me see./May I see?
c After
let we can put a phrase with a noun.
Let the person who made this mess clean it up.
Let the voters choose the government they want.
Let them decide.
Let them
decide means 'they should decide'.
NOTE
There are two special sentence patterns with a
similar meaning to the imperative. Both the subjunctive and may can express a wish.
God save the Queen.
May your dreams come true.
These patterns are rather formal and used only in limited contexts.