Legitimate means that someone is 'allowed' to participate in a community; for instance, someone who doesn't like social networks and thus refuses to sign up on dreamwidth can not become an experienced member of the dw community. What is perceived as legitimate or illegitimate depends pretty much on community rules and standards. Here at dw, pretty much anybody can sign up and start participating whereas in, say, a hospital certain requirements must be met first (you need to be a doctor, nurse etc.) before you're allowed to participate in hospital-related activities.

Peripheral participation simply means that someone who is new in a community and who doesn't know the ropes yet is located on the periphery of said community, because of limited knowledge and skills. That person's participation gradually increases in engagement and complexity until s/he, the former newcomer, has become a full part or member of the community. The focus in LLP is on informal learning structures and how knowledge and skills are developed in interaction with other, more experienced people.

Hope that answers your question :)