These rectangular 2x5 and 2x3 pin connectors seem to be commonly used for flat ribbon connectors. This example is from an AVRUSB programmer. What is the name of this style connector?
It's an insulation-displacement connector (IDC).
They're also known by a bunch of names IDC is common but also pretty broad. They're also sometimes called 100 mil header, or 0.100" header, 2.54mm header. Molex sells them under the CGRID III name. Samtec has a wide variety but here is a surface mount version.
They're so common that it's sometimes annoying to find the one you want since everyone has a brand name for them and they come in all different varieties, shrouded, unshrouded, locking, right angle, surface mount, through hole etc.
If you go here on digikey and select 0.100 under pitch you'll find more of them than you'll ever need.