What is the purpose of the weekly research meetings that advisers often have with their research group?
These meetings can serve a variety of purposes, but many of them fall into the broad category of keeping the adviser and all the group members updated on what everyone in the group is doing.  Below are some of the secondary benefits of a group meeting:


saves the adviser time (compared to meeting with students individually)
motivates the students to keep doing work, so they have something to report at the meeting
fosters a sense of community with the group
provides the students experience giving informal presentations
often leads to the more senior students mentoring the junior students


To answer your follow-up question in the description, no, some faculty meet with each student individually; in fact these separate meetings are common, for example, in mathematics.