Very simply, it is a loophole in Federal Labor Law that allows workers to take both Union and non-Union work.
The origin of the Financial Core statute starts with the National Labor Relations Act (NLRA) passed into law by Congress in 1935. This is the law that conferred upon private-sector employees the basic right to organize trade unions for the purpose of collective bargaining in order to ensure better wages and conditions in the workplace. The importance and legacy of this bill cannot be overstated. It established the Labor Unions as we know them today. Not only did this improve conditions for the union worker, but it also created an atmosphere in Industrial America that forced non-union job creators to compete for the best workers, therefore increasing wages and improved conditions outside the unions as well.
It also paved the way for the passage of the Fair Labor Standards Act (FLSA) in 1938 which established the Federal minimum wage, the 40-hour work week, overtime pay and stringent regulations for the employment of minors. Future amendments to the FLSA included The Equal Pay Act of 1963 and the Age Discrimination in Employment Act of 1967. It can be confidently asserted that without the unions and the NLRA, the FLSA and all of its important worker's rights protections would not have happened, or at least, not as soon.
However, as good as it is, there were a few deficiencies in the NLRA. One weakness, in particular, was addressed by the Supreme Court in 1963. Once a worker became a union member, said worker often ended up essentially a pawn of the unions which quickly became a powerful political and social force in America. For example, workers are often made to strike at a union's command even though they may be desperate to keep working to provide for their families. If they deviate from a strike order and cross a picket line, they do so at the risk of their union membership and potentially, their livelihood.
There are numerous other issues to cite: workers can be forced to slow down work on a project to pressure employers, whether they agree with the union's tactics or not; a portion of the worker's dues can be used to support organizations, politicians and causes which an individual employee might not favor; if no union jobs are available in a worker's field (within a reasonable distance from his/her home), the laborer must choose to either leave the union, go on unemployment, seek work outside of his/her trade, or take a non-union job- and risk being penalized. Penalties can include exorbitant fines, or even expulsion from the union, in which case the worker could not rejoin if a union job in his/her locale opened up.
In the Labor Board vs. General Motors case of 1963, The Supreme Court ruled to "limit the burdens of membership upon which employment may be conditioned, to the payment of initiation fees and monthly dues," by establishing the Financial Core statute. In the words of the court: "Membership, as a condition of employment, is whittled down to its financial core." In other words: If an employee in a union job refuses to respect any union-imposed obligations- other than the duty to pay dues and fees- and membership in the union would be otherwise denied or terminated, the condition of membership for purposes of employment (that is, the payment of dues) is nevertheless satisfied, and the employee may not be discharged from the union for non-compliance, but may opt to become a "dues-paying non-member" by fulfilling their "core financial obligation" to the union.
In 1988, the Supreme Court again addressed the Financial Core issue in the Communication Workers of America vs. Beck case. The question this time was whether an employee in a union job, who is a fi-core member, can be required to pay full union dues and fees if those fees are used for purposes beyond collective bargaining activities. The court ruled that the financial core obligation does not include "the obligation to support union activities beyond those germane to collective bargaining, contract administration, and grievance adjustment."
So, Financial Core is an option available to workers of any trade union which allows them to take non-Union employment if they either disagree with Union policies, or cannot find a Union job in their trade within a reasonable distance from their home. The FC option is anyone's legal right who chooses to exercise it. The Union is obligated to continue representing the Fi-Core status worker in all collective bargaining activities (this includes residual pay as well as health and pension benefits) but is not obligated to treat the "dues-paying non-member" as a full participant in good standing. The worker is obligated to continue paying dues, at a reduced rate, with the extraneous portion of dues (determined to be non-essential to collective bargaining) discounted. (For SAG-AFTRA, this discount is only about 4.5%) The worker can then take a Union job if it is available, as well as a non-Union job, without fear of being penalized or expelled for non-compliance.