SHORT TAKE:
Paterson is a charming film which follows an ordinary nice man for a week as he drives a bus and spends time with his wife and friends, finding inspiration in even the smallest things, to write poetry.
LONG TAKE:
(SOME SPOILERS)
Paterson spends a week in the life of a gentle, kind bus driver (Adam "Kylo Ren" Driver) in the small New Jersey town of the same name who lives with his 
artsy sweet and beautiful wife Laura and 
annoying bulldog Marvin.
The movie, to me, asks the question: do you affect art or does the art inherent in the creativity of those around you and the ambient beauty of everything from water falls to homeless bums to a 
pack of matches effect and shape YOU?
Paterson is presented as a very subtle fantasy – so subtle that I didn’t realize it until contemplating it after the credits had rolled. Paterson, the town, seems to be a magnet for creative forces which, in turn, effect their residents in large and small ways. This little unlikely town is home to a number of minor celebrities:
Lou Abbott – comedian, Patrick Warburton – actor, Victor Cruz – football player, Catherine Sullivan – astronaut, Andre Torres – baseball player.
And the film focuses on the unexpected artistry of Paterson, the man, a quiet government employee – a decent responsible man, faithful to and in love with his wife, 
observant and attentive to the needs of those around him, who finds enough beauty in even the most mundane detail of life – such as the name of the company on a box of matches – to inspire him to write poetry.
He and others seem at times almost under a spell which elicits bursts of creative energies.
But, I mean, why not? If spaceships like the Enterprise can be expected to attract temporal anomalies, and the house in Poltergeist be haunted by the angry spirits of unburied dead; if a fracture in time and space can be located in Cardiff, Wales from which Dr Who’s TARDIS can recharge; if demons can follow unwary owners of cursed objects; if Newton Haven in Simon Pegg’s The World’s End inexplicably can become the "shelter city" for evil alien robots who plan to replace humans; and the Darling Family attracts the attention of Peter Pan – then why can’t a town be imbued with its own creative forces and instill them in its inhabitants in one way or another?
An actor who dramatically obsesses over a childhood friend? 
A bar owner who strives to participate in chess competitions, even to stealing his wife’s Piggy Bank money? An adolescent girl who writes poetry waiting for her family in a back street in a style very similar to our protagonist?
Paterson – the bus driver – spends every week day waking up at 6:15, having a bowl of Cheerios, driving a bus through the sleepy community, listening to his passenger’s random chatter, spending his lunch at a water fall, 
enjoying his wife’s eccentric constant redecorating and cooking, taking his wife’s dog for a walk, having one beer at a local bar and entertaining himself all day, like a familiar tune he hums constantly, writing strains of free style poetry in his head then committing them to a solitary, uncopied notebook which he seems interested in only his wife being privy.
Elements of his wife’s morning-described dreams faithfully and routinely crop up in his every day life – she mentions having seen twins and suddenly Paterson notices they are everywhere.
Opportunities for poetic events gently flitter around him like fairies. And people simply act in prosaic but poetic ways (sounds like an oxymoron but it works here): The bartender plays chess with himself then moans about getting his "ass whooped" by his opponent. The smitten lover brings a toy gun with which to confront his ex-girlfriend. 
A random Japanese tourist commits a random act of kindness which gets our protagonist back on track after a minor catastrophe. His Iranian born wife suddenly announces her "lifelong dream" of which our protagonist does not appear to have had any foreknowledge, of being a country western star, and spends her days painting different black and white patterns on everything that stands still long enough – from the curtains to the dog’s collar to the cupcakes she sells at the Farmer’s Market.
And our bus driver is a poet who aspires to be…
a bus driving poet happily married to his beautiful doting wife.
Paterson is a charming little movie worth seeing as one might meditate on the shape of clouds on a still summer afternoon or watch swans swim on a glass flat lake.
And those who only know Adam Driver as this generation's version of Darth Vader should watch to note that Driver really can act.