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"Let's" is commonly used in modern English when you say:
Let's do something...
eg. Let's...play football, go to the cinema, have a pint down the pub, etc.
No one uses the original "Let us" form in this context anymore unless they are aristocracy!
eg. "I say old chap - let us go boating upon the river this afternoon what what!"
"Let us" is still used but in a different context and to mean "allow/permit us".
eg. allow us...to do something
...Let us in (to the house), Let us help you, etc.
let's and let us 
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