With examples from "X-Files"
It’s time for another Blast from the Past, and this time we’re going to rediscover the difference between let’s vs. lets. mab_browne already did a great job of covering this topic, so I'll just do a recap for you.
Let's start with the definitions. Lets is defined as "allows" or "permits." It's a verb for "to let" or "to allow" and is primarily used when speaking of only one person, but occasionally useful when speaking of a group.
Mulder knows the real truth, but he still lets Scully try to convince him otherwise. Her reasoning sometimes helps them figure out where to go next.
Darkness is only a deterrent, not really a danger for Mulder and Dana, but she lets him go first anyway.
Basically, if you can substitute "allow" in the sentence, use lets.
Let's is simply a contraction for "let us" and is almost always used in dialogue.
"Let's go into the church," Mulder said in a tone meant to end the discussion of whether or not ghosts really did exist. "We find nothing, we go. But we should at least check it out."
"Mulder, let's come back at a later time," Scully said after a particularly huge piece of altar barely missed hitting her in the head.
"Yeah," he muttered softly as he stared at the manifest bearing down on him. "Let's just get out of here."
Just remember, if you are able to substitute the word "allow" in the sentence, use lets, and if you are writing about more than one person wanting to do something together, use let's.