Basically, in the time it took to wait for an adoptive cat to be ready to take home (30 days), they freaked out about committing to have a pet and where they are in their lives. They break up because Jason changes jobs to be happier, and Sophie sleeps with a weird guy. The cat gets put down because they forget about her, because both of them were too involved in themselves. It's unclear if they ultimately reconcile with each other.
Sophie (Miranda July) and Jason (Hamish Linklater) are a married couple who are very poor and kind of stuck. (Sophie teaches at a dance school for kids and is struggling to do even one dance for a "dance a day" youtube video project all her coworkers are doing and Jason does tech support from home so he's always around the house on the computer.) They decide to adopt a cat at the local shelter.
While they wait, Jason purchases a painting for charity. The little girl whose dad did it tells him that her dad and mom's phone numbers are written on the back in case the person who buys it decides that he/she doesn't like it.
The kitty they've decided on is pretty sick. We see the cat's paws periodically, one wrapped up in a bandage and one free. They name the cat "Paw Paw" but will have to wait 30 days till they can come back and get it. Periodically we see Paw Paw and she talks about feeling that she's their cat, looking forward to not living outside and having a home, and how difficult it is waiting till they come for her.
In the mean time, Jason and Sophie go home and discuss how they haven't achieved what they wanted to, that the possibility of this cat living a longer than expected time means they need to get the fun stuff done before they "commit" to having a pet.
Jason quits his job and starts working for a eco-group selling trees door to door and this makes him happy. While he's doing this he has weirdly personal conversations with a few men, in particular, about relationships and one man shows him a collection of many cards/poems/stuff he made for his wife over their long marriage and tells Jason that it helped them through the hard stuff. He tells Jason that his relationship (4 years together) is just in the beginning stages.
Sophie is frustrated about the dance thing, is very isolated (having become to embarrassed to email her friends since she hasn't done the dance-a-day thing she told them she was going to do) and acts like she's unhappy that the tree-sale thing is making Jason happy so she calls the dad's number on the back of the kid painting. They have odd conversations about where they are in the city, then she meets him to have a sign made (because that's his business), and eventually sleeps with him.
She confesses to Jason in the middle of the night and kneels on the floor. He puts his hand on her head and then has a conversation with the moon, voiced by the old guy who made the cards for his wife, about how he wants to hold off the next moment. The moon informs him that they've been in that position for days and that whatever's going to happen the next moment he has to let go of her head.
We see Paw Paw one last time and she tells us that she was so looking forward to being their cat that when she dies she can't leave her cage because she's still waiting for them.
Sophie moves in with the guy and his daughter (it's a little unclear when that happens) and they kind of become an awkward family. She helps the daughter dig a hole in the back yard and then the dad buries his daughter up to her neck overnight. She does a weird dance with a t shirt she loves (I think it belongs to Jason) in the living room and then the dad throws the t shirt in the garbage.
Paw Paw's empty cage is seen again and she talks about being part of the light and no longer being their cat, a cat, or even an "I" anymore, just being part of the light so we know she's in better place.
Jason and Sophie then both realize they've forgotten the cat. The shelter tells them separately that Paw Paw has died, that they waited an extra day. Jason has an awkward conversation with a man he's trying to sell a tree to and then Sophie comes home and they have a conversation and he tells her she can stay - but just one night. She puts things in a suitcase in her pjs and he sits on the couch reading and the credits start.