What does Bernard mean?
From Germanic 'bern' (bear) and 'hard' (brave, hardy), 'brave/strong as a bear' - the bear being a warrior symbol. Borne by several saints.
What does Reginald mean?
From Germanic 'ragin' ('counsel, advice') + 'wald' ('rule, power'), i.e. 'ruler by counsel' / 'counsel-power'. Note: this is 'ruler/power', NOT literally 'king'. Reynold and Ronald are closely related forms.