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Function

body& or Function[body] is a pure (or "anonymous") function. The formal parameters are # (or #1), #2, etc. 

x|->body or xbody or Function[x,body] is a pure function with a single formal parameter x. 

{x1,x2,…}|->body or {x1,x2,…}body or Function[{x1,x2,…},body] is a pure function with a list of formal parameters. 

Function[params,body,attrs] is a pure function that is treated as having attributes attrs for purposes of evaluation.

Please visit the official Wolfram Language Reference for more details and examples on core symbols.