What do you call the kind of female header found on Arduinos and shields?
They go under a variety of names. These connectors are usually listed in a category called "Rectangular Connectors - Headers, Receptacles, Female Sockets" (205377 items on Digi-Key)" Then you specify number of rows, pitch size (2.54mm or 0.100"), etc. 3M calls them "boardmount sockets". They are also known as "female headers" (Sullins), or "vertical receptacle" by Molex and FCI-Amphenol. Or just simply "0.100" receptacle".

The cost of headers varies depending on complexity of pin receptacle construction (from simple fork-like receptacle, through two-four-sided leaf springs, to CNC machined round ones with multiple tiny springy wires inside). The next level of pricing depends on plating and its thickness, from tin-nickel on cheap side, to gold, from 0.05 um finish to 5 um finish. All this results in different contact current rating.