Recommendations for Few Tables/Small Space/Intimate
you have to try tia pol! they have super yummy tapas, and the space is very cozy. it's a narrow brick-walled restaurant with high dark wood tables lined with benches or high-backed stools. the sangria is great, and i've loved everything on the menu that i've tried. i used to live around there, and went to tia pol on a regular basis. it's on 10th avenue between 22nd and 23rd.
for pretty good and cheap italian around there, pepe giallo is also an intimate space - on 10th avenue between 23rd and 24th.
in west village:
blue ribbon bakery (upstairs)
those places come to mind. there are tons of cozy spaces in west village, though. have fun exploring!