The words let's, lets sound the same but have different meanings and spellings. Why do let's, lets sound the same even though they are completely different words?
The answer is simple: let's, lets are homophones of the English language.
To give permission or opportunity to; allow: I let them borrow the car. The inheritance let us finally buy a house. See Usage Note at leave1.
To cause to; make: Let the news be known.
Used as an auxiliary in the imperative to express a command, request, or proposal: Let's finish the job! Let x equal y.
Used as an auxiliary in the imperative to express a warning or threat: Just let her try!
To give permission or opportunity to; allow: I let them borrow the car. The inheritance let us finally buy a house. See Usage Note at leave1.
To cause to; make: Let the news be known.
Used as an auxiliary in the imperative to express a command, request, or proposal: Let's finish the job! Let x equal y.
Used as an auxiliary in the imperative to express a warning or threat: Just let her try!
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").