"He lives here? Here? In this Muggle dunghill? We must be the first of our kind ever to set foot--"
-- Bellatrix Lestrange (HBP2)
“I know who you are. You’re that Snape boy! They live down Spinner’s End by the river....Why have you been spying on us?”
-- Petunia Evans as a child (DH33)
Spinner’s End is a street in Cokeworth, a dark industrial Muggle town, most likely somewhere in northern industrial England. It is on this street that Snape’s house sits, and down which Narcissa Malfoy ran – with Bellatrix Lestrange in pursuit – to seek him out HBP2. Somewhere nearby, there is also a Muggle playground where a young Snape first met Lily and Petunia Evans, and most likely the house they grew up in as well (DH33).
Severus, Lily and Petunia grew up together in the town of Cokeworth, which is where Spinner's End is located (Pm)
During Occlumency lessons, Harry used a Shield Charm against Snape's Legilimens and accidentally saw some of Snape's memories that took place at Spinner's End: "...a hook-nosed man was shouting at a cowering woman, while a small dark-haired boy cried in a corner… a greasy-haired teenager sat alone in a dark bedroom, pointing his wand at the ceiling, shooting down flies…" (OP26)
Spinner's End is near a "dirty river" and overshadowed by a huge chimney left over some from forgotten mill. Bellatrix Lestrange refers to it as a "Muggle dunghill" (HBP2)
The name "Spinner" probably refers to the cotton-spinning industry in the north of England. In 1912, the British were making 8 billion yards of cotton cloth a year, but as other countries industrialized over time and India boycotted British-made textiles in favor of their own, the cotton mills began to close down one by one.
The 19th-century novel North and South by Elizabeth Gaskell is set against the backdrop of a harsh cotton mill town called Milton, which the author based on Manchester. In the novel, the main character Margaret Hale describes the town and the effect of all the cotton fibers polluting the air, which sounds very much like the Dementor's mist hanging over Spinner's End:
"... outside a thick fog crept up to the very windows, and was driven in to every open door in choking white wreaths of unwholesome mist."
J.K. Rowling lived in Manchester at one time, and it was on the Manchester to London train that she first conceived the story of Harry Potter. It's quite possible that Cokeworth was also inspired by Manchester (JKR).
For a detailed explanation of its probable location, see the essay, "Where is Spinner's End?" by Claire M. Jordan.