Solvents are added to oil paints for temporary change, they are designed to work in a way such that they evaporate consistently and completely as the oil paint dries. (Technically, the more correct term is diluents, as not all are solvents, but it's not the term very commonly used.) Solvents are also used to soften resins, making mediums, cleaning up, and also for cleaning brushes. It is very essential to use solvents in a well-ventilated room and you need to remember that they are flammable (catch fire easily).
Turpentine is the traditional solvent used in the oil painting. It's actually based on tree resin and has a very fast evaporation rate, releasing harmful vapors. It could also be absorbed through healthy skin. Try using only artist quality turpentine as the industrial variety you will find in hardware stores probably contains impurities; it needs to be colorless, like water. It is also known as spirit of turpentine, oil of turpentine, genuine turpentine, English turpentine, distilled turpentine; double rectified turpentine, or even simply turps.
Mineral spirits is based on pure petroleum and has a moderate evaporation rate, releasing harmful vapors. It is not absorbed through healthy skin like turpentine. Mineral spirits is less expensive than turpentine. Mineral spirits is a stronger solvent than odorless mineral spirits, also called as white spirits.
Odorless mineral spirits is based on petroleum and has a reasonable evaporation rate. It is again not absorbed through healthy skin. It's quite expensive than normal mineral spirits as it has had some of the harmful perfumed solvents removed. Good brands include Turpenoid, Thin-ex, and Gamsol.
Tip: Test the quality of a solvent by putting a little on a drop of paper and to let it evaporate. If it doesn't leave any resident, stain, or smell, it needs to be good enough for oil painting.
Resins are used to increase the shine of oil paint, reduce the color, reduce drying time of a medium, and add body to drying oils. The most used one is a natural resin known as Damar that should be mixed with turpentine as it would not thoroughly dissolve when mixed with mineral spirits. Damar could also be used as a varnish.