Eating stalker meat will stun the player and turn them temporarily invisible for 50-149 turns if they are not already invisible; if eaten while already invisible, the player will become permanently invisible and gain the see invisible intrinsic. Wearing a mummy wrapping counts as being visible for this purpose.
Players polymorphed into stalkers are permanently stunned.
Stalkers are always generated invisible as indicated above, and will always be hostile when generated normally.
Stalkers can hit fairly hard and move at the same speed as a normal player, making them annoying to face or avoid unless you have a means of detecting them such as telepathy.
Stalker meat is one of the more reliable sources of the see invisible intrinsic, either through eating two separate servings in quick succession or else eating one with a wearable source of invisibility. As eating stalkers stuns you, be sure to avoid the temptation to do so in the midst of battle unless you can cure it quickly (e.g., with a unicorn horn).
The stalker is based on a Dungeons & Dragons monster known formally as the invisible stalker, which is a type of air elemental in D&D. This is the origin of the E monster symbol, even though stalkers are nothing like air elementals in NetHack.
As most versions of Hack and NetHack automatically prefix monster names with "invisible" when relaying actions referring to an invisible monster, the word "invisible" is dropped from the monster's name to avoid the redundant "invisible invisible stalker".
The symbol I was used to represent the stalker (I meaning "invisible") until NetHack 3.3.0, where it was appropriated to indicate the last known position of any invisible monster.