Babby is an Internet-speak corruption of "baby." It is common in the US for toys to advertise themselves as "baby's first $ITEM", such as "baby's first spatula" or "baby's first hammer."

Thus, the project being discussed is a toy attempt at building a web server. This is a fairly common mid-sized project for learning a new language, as it implicates lots of the stdlib (file I/O, networking, etc), has a defined scope with lots of opportunity for flourishes, and is immediately showable.