Suppose I have a table that stores a house’s latitude and longitude.
If a house’s lat/long puts it within a certain boundaries, which would be specified by a polygon made up of other coordinates, I’d like to modify the house’s “Neighborhood” column by whatever the corresponding neighborhood value is.
What’s the logic to use latitude and longitude to determine if an object is within a certain set of boundary criteria?
This depends on the complexity of your polygon.
Here is a good answer which covers the possibilities
And off to Null Island you go! 
(For those that dont get the reference, “Null Island” is a fictitious island at 0,0 on the lat/long plane; it in fact contains nothing but ocean water and a weather buoy, as those coordinates are in international water off the coast of Africa, and thus have no houses there, which makes it a good anchoring point for ray calculations [not to mention making the maths easier]).