The Rug (Symbol)
The rug is central to the plot, because after Treehorn's guy urinates on it, the Dude goes to Lebowski to get compensated for it, which sets the whole plot in motion. While the Dude does not seem to care about much in his life—he goes to the supermarket in a robe, is unemployed, and is almost permanently stoned—he is uncharacteristically intent on seeking reparations for his damaged rug. The rug comes to represent the Dude's self-respect, his ability to advocate for his own needs in society. It is not necessarily the rug that is important to the Dude, but the fact that he was disrespected and his property was treated with callous disregard.
Tumbleweed (Symbol)
In the beginning, we see a tumbleweed rolling across the landscape, through various Los Angeles locations as "Tumbling Tumbleweeds" plays. The tumbleweed represents not only the fact that the movie takes place in the West, a symbol of old Wild West movies and the American cowboy, but also of the drifting and aimless nature of the Dude and his story. The tumbleweed is a fixture of the landscape, something which is always moving, but has no agency, in thrall to the wind. Thus, the tumbleweed represents the attitudes of the Dude, who also goes with the flow to such an extent that he hardly ever has any control over his own destiny.
Toe (Symbol)
Jeffrey Lebowski shows The Dude a toe sent to him by the kidnappers. It has the same nail polish that Bunny was wearing the last time the Dude saw her, on their first meeting. The toe, revealed later to be someone else's, is a symbol of the danger that Bunny is in, and the fact that the stakes are higher than they once were. The toe symbolizes the grotesque and comic brutality of the mystery of Bunny's disappearance.
Bowling (Motif)
The Dude does not have a job, but he is committed to his hobby of playing in a local bowling league. He and his friends Walter and Donny take bowling very seriously, and the bowling alley becomes a motivic setting in the film. It is both a place of competition and tension—as we see the friends fighting with their bowling opponents—and a place where the world is suspended, a place that exists outside of time in the same way the Dude exists outside of time.
"Sometimes you eat the bear, and sometimes the bear eats you" (Allegory)
This is something that the Stranger says to Dude while they sit at the bar at the bowling alley. It is a strange koan, a philosophical idea that is fairly straightforward. In his estimation, life can be likened to a man in the wilderness getting into a conflict with a bear. Either the man is victorious over the bear or the bear is victorious over the man. The story is an allegory for the unpredictability of life, the fact that life is made up of different conflicts with varying results.
