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1.
To chicken out or back away from something or someone.
As from the movie "Reservoir Dogs":

"Don't pussy out on me now, they believe you, they believe every fucking word you're saying. And you know why? Because you're super cool."
by Freddy February 02, 2004
 
2.
To back out of everything like a little girl, especially fights.
Al Ventry likes to pussy out of fights.
by Da Finest!!!! May 06, 2004
 
3.
Relating to the word "Pusillanimous", it means to lack courage or resolution. When someone "pussies out" it means they have no backbone, or lack the courage to do something.

Pussy.
"Hey man, you should come chill with us later and take some ecstasy!"
"Sorry, but I don't want to lose an ice cream scoop of my brain cells, thank you very much."
"C'mon man, don't pussy out on me."
by seplion456 February 23, 2010