Typically every professions from every sectors (employees, medical, farmer, etc.) end up into a given category (social workers, railroad workers, etc.). Unions then use the memberships of all the employees across different companies and sectors as a leverage to negotiate better working conditions or retirement on behalf of their affiliates. So when/if is a strike is called for for a particular problem at company X or in sector Y they get the full force of every affiliates behind the union (edit: they can also dispatch strikers from company X at company Y location so workers from company Y aren't directly threatened or put into uncomfortable situations for instance, and massive street demonstration can rally many workers from many different companies/sectors).
Of course that system has been built over decades and the way it operates and negotiations are handled can vastly differ between neighboring countries (think France which is more confrontational at first and Germany where unions are more integrated into the process) (plus, unions have different political outlooks, so it's not one size fits all).
In countries with a strong social net unions use their affiliate's entrance fees to guarantee they are paid for striked days. Also some countries have laws that forbid employers to punish strikers (granted they announce in time they are going on a strike). Some sectors (medical, police, etc.) have some restrictions on that "right to strike".
That's just a really short resume, though and I don't know if such a thing could be possible in the USA.