The aim of the present paper is to extend the concept of a congruence from lattices to posets. We use an approach different from that used by the first author and V. Snášel. By using our definition we show that congruence classes are convex. If the poset in question satisfies the Ascending Chain Condition as well as the Descending Chain Condition, then these classes turn out to be intervals. If the poset has a top element 1 then the 1-class of every congruence is a so-called strong filter. We study congruences on relatively pseudocomplemented posets which form a formalization of intuitionistic logic. For such posets we define so-called deductive systems and we show how they are connected with congruence kernels. We prove that every strong filter F of a relatively pseudocomplemented poset induces a congruence having F as its kernel. Finally, we consider Boolean posets which form a natural generalization of Boolean algebras. We show that congruences on Boolean posets in general do not share properties known from Boolean algebras, but congruence kernels of Boolean posets still have some interesting properties.