The term corporatism is used refer to two totally different concepts:
1. A social theory about social-economic relations;
2. The influence of businesses in politics.
When we use corporatism on this site, we always refer to the first concept. The second concept is more accurately referred to as corporatocracy (= rule by businesses). On this second meaning we can be short: corporatocracy is incompatible with republicanism, as res publica means the general interest and hence a republican government furthers general interests. On contrast a corporatocracy furthers the private interests of certain businesses over the interests of the public at large.
Corporatism (in the first sense) is a more interesting concept. In order to understand this school of thought, we have to know that it was as a reaction to Marxism. According to Marxists there is an irreconcilable opposition between capitalists (= business owners) and workers: the latter are exploited by the former. They believe that the only way this conflict can be resolved is by a (violent) revolution during which the capitalists are expropriated.
With the horrors of the French revolution in mind, corporatism was developed, first by the Roman Catholic Church but later also adopted by secular groups. Corporatists reject the notion of a revolution to improve the conditions of workers, instead they promote regular negotiations between employers and workers (usually represented by their trade unions). For this purpose each sector of the economy is organized into corporations (hence the name). In each corporation representatives of employers and employees will meet at regular intervals to discuss various topics ranging from wage to number of holidays to workplace safety.
Since the agreements made in a corporation are binding to all people in that sector, there is for no need for minimum wage laws in such a system. For instance there is no legal minimum wage in Sweden. In that country over seventy percent of the labour force is member of a trade union, and minimum wages are set by collective bargaining.
There are several variants of corporatism, though all of them include employers and workers. The most common varieties promote tripartite corporation: with additionally representatives of the state. Here we have to distinguish between fascist  and democratic corporatism. In the former variety representatives of the fascist party were sitting besides employers and workers. Unsurprisingly, this was actually more a tool control the economy rather than an honest attempt to improve the conditions of workers.
Democratic corporatism is the combination of corporatism with parliamentary democracy, and was the norm in West-Europe in the decades after the second world war. In this variant representatives of a democratically elected government would meet with representatives of employers and workers.
The rationale of tripartite corporatism, and the democratic version in particular, is to ensure that sector interests would not infringe on public interests. The public interests would be defended by the government’s representatives.
One of the mistaken assumptions of Marxists is that only classes can have shared interest, but that capitalists and workers cannot have a shared interest. However, they ignore the importance of sectors. In reality capitalists and workers in a certain sector can have a common interests, which would put them in conflict with capitalists and workers in other sectors.
In early post-war Europe corporatism was essentially the social economic paradigm of christian-democrat and social democrat parties. The difference between social- and christian-democrats were primarily found in non-economic issues such as contraceptives, abortion, homosexuality and so on.
For further reading:
 We mean fascism in a historical sense, i.e. the movement of Benito Mussolini and the movements directly inspired by him during the same period. Unfortunately there is currently a tendency to call all authoritarian movements fascist with no regard of history.