In the show Will is the deserter of the Night's Watch that gets executed by Eddard Stark after he is captured. He is captured in the hills and the Starks ride out to meet the deserter somewhere in the hills.
Rodrick Cassel: Lord Stark. My lady. A guardsman just rode in from the hills. They’ve captured a deserter from the Night’s Watch.
Eddard Stark: Get the lads to saddle their horses.
Game of Thrones, Season 1 Episode 1, "Winter is Coming"
In the scene in which Will is executed the place seems to be for execution specifically. There is a mound in the middle of a clearing with a log on it that has clearly been used before. Obviously they could have brought the log with them for the purpose of the execution but this seems less likely in my mind. Especially as the Starks rode out of Winterfell to the deserter rather than the deserter coming to them.
What is this place where the execution takes place?
It's worth noting that this happens differently in the books so I will be looking for show only answers. In the books Gared is the ranger who deserts and is executed. He is then captured in a small holdfast and kept there until the Starks arrive and execute him there.
The man had been taken outside a small holdfast in the hills. Robb thought he was a wildling, his sword sworn to Mance Rayder, the King-beyond-the-Wall. It made Bran's skin prickle to think of it. He remembered the hearth tales Old Nan told them. The wildlings were cruel men, she said, slavers and slayers and thieves. They consorted with giants and ghouls, stole girl children in the dead of night, and drank blood from polished horns. And their women lay with the Others in the Long Night to sire terrible half-human children.
But the man they found bound hand and foot to the holdfast wall awaiting the king's justice was old and scrawny, not much taller than Robb. He had lost both ears and a finger to frostbite, and he dressed all in black, the same as a brother of the Night's Watch, except that his furs were ragged and greasy.
A Game of Thrones, Bran I