Annealing, in a general sense, refers to treatments that reduce the hardness of steel or create a structure that facilitates subsequent treatments, both cutting and plastic. When only the word annealing is used, it often refers to soft annealing. For the sake of clarity, the type of annealing should always be specified.

Annealing operations are included in the manufacturing chain to give the steel a microstructure that facilitates further processing and provides predictable strength properties. Many annealing operations are not performed by the end user, but by the steel producer. Examples include annealing of semi-finished products such as strip, bar, tube, forgings and cast blanks.
Ref. Steel and Heat Treatment – A Handbook (8.2)

The purpose of stress-relief annealing is to reduce or eliminate unwanted residual stresses, which have built up in the material during both machining and plastic deformation and which could cause deformation during subsequent heat treatment or during the component’s use. Stress-relief annealing is performed on welded parts and structures and on cold-formed or machined parts. Stress-relief annealing is performed at a lower temperature than recrystallization annealing.
Ref. Steel and Heat Treatment – A Handbook (8.2.5)

Soft annealing is the method that makes the steel as soft as possible and is performed on tool steel and low-alloy steels with high carbon contents to facilitate machining with cutting tools.
Ref. Steel and Heat Treatment – A Handbook (8.2.2)

Solution annealing is used for stainless austenitic steels and involves homogenization annealing or, in some cases, recrystallization annealing. With slow cooling after (heating), secondary phase particles are precipitated in the austenitic steels. The precipitation usually occurs in the grain boundaries. For stainless, austenitic steels, this means a reduction in the chromium content near the grain boundaries, which impairs the corrosion resistance locally and can result in intergranular corrosion (grain boundary corrosion). The precipitation can also occur through (heating) in the temperature range 500-850°C.

Solution annealing is performed to avoid the problems that the secondary phase particles can cause. Solution annealing involves (heating) to a temperature where the austenite is homogeneous, usually 1000-1100°C, followed by rapid cooling, for example in water. For small dimensions, air cooling may be sufficient. Deep-drawn sheet metal parts with material thicknesses up to 4 – 5 mm are often solution annealed in a vacuum furnace with forced gas cooling.

The matrix in the austenitic steels does not undergo any phase transformation that gives grain refinement during (heating) or (cooling). During solution annealing, grain coarsening can instead be expected, but it normally has no significant impact on the toughness. Austenitic steels that are cold-worked recrystallize during solution annealing, which leads to a decrease in hardness and an increase in ductility. With certain combinations of reduction degree, temperature and time, for example just below the critical degree of deformation, abnormal grain growth can occur.
Ref. Steel and Heat Treatment – A Handbook (8.2.6)

When cold working steel, the hardness increases through work hardening, and the possibilities for continued plastic working decrease. Recrystallization annealing is therefore performed between the cold deformation steps during cold rolling of strip, cold drawing of wire, deep drawing of sheet metal or after other plastic forming, which is so severe that continued plastic working is difficult.

During annealing above approximately 600°C, the steel recrystallizes, which means that new, stress-free, undeformed grains nucleate and grow, at the expense of the deformed grains. The microstructure is often more fine-grained than the original one. Due to work hardening, the hardness is high before recrystallization, whereby it decreases to the hardness of the original, undeformed structure, when new grains have formed after recrystallization.
Ref. Steel and Heat Treatment – A Handbook (8.2.4)