The book I read to research this post was Light & Shoot: 50 Fashion Photos by Chris Gatcum which is an excellent book which I bought from amazon.
There are 3 types of flash lighting incandescent which is the cheapest but produces heat & doesn’t mix well with daylight. HMI is more expensive & may have to be rented in a lot of cases but is cool running & mixes with daylight. Fluorescent is a kind of half way although the cheaper ones may not be daylight balanced. A shield called the barndoors shields portions of the studio from the light. A beauty dish softens the light. A boom arm positions the lighting. A fill light fills in shadows. A flag is similiar to the barndoors. A flashmeter measures the amount of light hitting a certain spot. A fresnel or spot is a small focused light. A grid diffuses and/or adds a texture to the light. A light stand is similiar to a boom arm. A radio slave controls the flash from the camera. A slave cell controls the flash from another flash & will work in conjunction with a radio slave. Finally an umbrella reflects & diffuses the lighting.