According to USCharterSchools.org, "[c]harter schools are nonsectarian public schools of choice that operate with freedom from many of the regulations that apply to traditional public schools. The 'charter' establishing each such school is a performance contract detailing the school's mission, program, goals, students served, methods of assessment, and ways to measure success."
Within LAUSD, there are two types of charter schools - a "conversion charter" whereby an existing public school is converted to a charter school, and a "start-up charter" established by a member of the public. Some charter schools are affiliated with LAUSD, while others are independent.
District-affiliated charter schools adhere to all LAUSD guidelines and policies but have more autonomy with their school budget and curriculum choices than traditional schools. They can purchase services from LAUSD, hire LAUSD teachers who remain part of the teachers' union, and have access to free LAUSD facilities.
Independent charters have even greater autonomy than district-affiliated charters but receive no organizational support from LAUSD, meaning they must handle their own hiring, employee benefits, payroll, facilities management and more.
Charter schools are public schools. There is no formal tuition, but most schools fund raise and solicit donations. (I know of one school that asked parents to give over $1,000 last school year.) Many schools also have substantial parent participation requirements.
There are over 180 charter schools within LAUSD. You can find a list of charter schools here.