This habilitation thesis fits in the fields of algebraic and enumerative combinatorics, with connections with computer science. The main ideas developed in this work consist in endowing combinatorial objects (words, permutations, trees, integer partitions, Young tableaux, etc.) with operations in order to construct algebraic structures. This process allows, by studying algebraically the structures thus obtained (changes of bases, generating sets, presentations by generators and relations, morphisms, representations), to collect combinatorial information about the underlying objects. The algebraic structures the most encountered here are magmas, posets, associative algebras, dendriform algebras, Hopf bialgebras, operads, and pros. This work explores the aforementioned research direction and provides many (functorial or not) constructions having the particularity to build algebraic structures on combinatorial objects. We develop for instance a functor from nonsymmetric colored operads to nonsymmetric operads, from monoids to operads, from unitary magmas to nonsymmetric operads, from finite posets to nonsymmetric operads, from stiff pros to Hopf bialgebras, and from precompositions to nonsymmetric operads. These constructions bring alternative ways to describe already known structures and provide new ones, as for instance, some of the deformations of the noncommutative Faà di Bruno Hopf bialgebra of Foissy and a generalization of the dendriform operad of Loday. We also use algebraic structures to obtain enumerative results. In particular, nonsymmetric colored operads are promising devices to define formal series generalizing the usual ones. These series come with several products (for instance a pre-Lie product, an associative product, and their Kleene stars) enriching the usual ones on classical power series. This provides a framework and a toolbox to strike combinatorial questions in an original way. The text is organized as follows. The first two chapters pose the elementary notions of combinatorics and algebraic combinatorics used in the whole work. The last ten chapters contain our original research results fitting the context presented above.