Peuple is only used when speaking of the individuals of a nation. For example, the English people : le peuple anglais.

Gens is used when speaking about individuals without connection between them. You could replace it by "ils" ou "on". For example, les gens mangent très mal : people eat very badly.

Personnes is used when talking about individuals. For example, les coureurs sont des personnes saines : runners are healthy persons.

Hope it can help.

Well, I find you something with a great explanation.

"les gens" is a general term who generaly mean "everybody but me". 

"personne" is a paradoxal word because it is both undefined and precise. If you say "cette personne" you are targeting an unique being without actually talking of it. Moreover "personne" can also mean nobody.