Let x Lets x Let's

According to the Macmillan Dictionary :

Let ("lets" is the third-person ) - "to allow something to happen" :

I stepped back and let him pass.

Here, let me help you. 

She lets her hair down (Tokens' song) 

Let's - "the usual way of saying of writting 'let us' ". Use it when you are making a suggestion:

Let's go, we are late.

Let's not talk about it. 

Now, let me recommend an excellent youtuber English teacher, who has a video talking about this: