Perception determines how far away enemies can be detected on the compass (enemies show as red bars). When playing with a high Perception, it is possible to "sense" the enemies even before they come into view.
Contrary to what the name would make one believe, Perception does not actually increase V.A.T.S or regular accuracy at all.
If Perception is at the most extreme (highest or lowest), Doc Mitchell will either respond that nothing gets by the Courier, thinking he could use them when he lost his keys last month [of awakening the Courier] (for highest) or that he is no optometrist, but suggests the Courier should be fitted some glasses while they're at the Doc's place (For lowest).
Dialog with Christine has numerous Perception checks between 6 and 7. Intelligence checks of similar levels can substitute for some, but not all of them.
It is possible to artificially get a Perception of 10 with only 6 points at the start of New Vegas by combining the Four Eyes trait, Lucky Shades, a hat and the Perception Implant (6 - 1 + 3 + 1 + 1 = 10).
Similarly, it is possible to achieve a temporary boost to 10 from a mere score of 2 using both kinds of Mentats, Absinthe, Coyote Tobacco Chew, and a hat (2 + 4 + 3 + 1 = 10).
By having access to DLC items, it is possible to get up to PE 10 without the Four Eyes trait, even if starting at only 1. The Elite riot gear helmet (+2) can be worn together with the 1st Recon beret (+1) and a Stealth suit Mk II (+1). After installing the Perception implant, Perception is at 6. From there, any combination of Perception-enhancing chems can reach 10.