Bourré is the past participle of bourrer, which means “to stuff”. It can mean “filled to the very limit”.
J’ai pris le métro à Châtelet, il était bourré à craquer.
I took the metro in the Châtelet station, it was completely full (“so full it was on the verge of bursting”).
Ce chanteur est bourré de talent.
This singer is full of talent.
However, this word has another common, colloquial meaning: “drunk”. In spoken French, it is much more common than the official words for “drunk”, ivre and saoul (alternatively spelled soûl and pronounced sou in both cases).
Faut vraiment être débile pour conduire bourré.
You really have to be stupid to drive when your drunk.
T’es vraiment lourd quand t’es bourré.
You’re really annoying when you’re drunk.
J’étais bourré, je me souviens plus de rien.
I was drunk, I can’t remember anything.
Listen to the examples: