A public law institution is any body constituted to exercise or oversee the use of any form of public power. So, in the previous example, a bank run under the auspices of the government qualifies, as would a national postal service or something similar. More usually, when speaking of public law institutions, you would be referring to the tribunals and checking mechanisms operating under administrative law. This would include ombudsmen, or refugee tribunals; less formal venues than courts, but providing a review of administrative action through examination of the merits of a case (as opposed to pure legality).
This'll all depend on the country you're in, though.