Steel is an alloy of iron combined with a small amount of carbon. It is used in watchmaking for a wide variety of parts ranging from simple screws, to watch cases, to more complicated parts such as pinions or escapement wheels. Steel can be modified and made harder or softer by heat treatment. Its alloy can also be modified to make it stainless or non-allergenic.
Steel can be used for many parts of the watch. For the casing, i.e. the external parts of the watch, stainless steel is obviously used. For the parts inside the movement, other more important characteristics such as mechanical resistance are preferred. For a long time and sometimes still today, the parts inside the watches were made of oxidizable steel, which could rust. This is why watchmakers historically never touched these parts with their hands, as the sweat from their hands would rust them. Even for large parts made of oxidizable steel, the qualified watchmaker always uses tweezers to handle them.
Steel has the advantage of being a material whose characteristics can be modified by heat treatments. Annealing can make it softer or hardening can make it harder. If you want to have a spring steel, it will be a mixture of both. When heated, it also changes its color by surface oxidation which remains modified when returned to room temperature. This is how blue steel needles are made, for example, by heating the steel to 300 degrees Celsius.
Steel, when it is not alloyed with other metals to make it stainless, has the defect of being oxidizable. Like all ferrous metals, this oxidation on this metal has the defect of going in depth and oxidizes, therefore rusts, the part in depth. Steel is also a rather hard metal and not so easily machinable, which is why it is almost never used for a watch plate and also rarely for bridges.