Let $\{A_i\}_{i\in I}$ be a family of $k$-algebras, where $k$ is a field and $I$ is a partially ordered set. Does the following colimit exists? $$ \operatorname{colim}_{i\in I} A_i $$ How can it be described? (I mean like an analogue for (co)-limits of modules)
What about the limit?
In general, all limits and colimits of $k$-algebras exist. Limits are taken set-theoretically. To describe colimits it suffices to describe coproducts and coequalizers. The coproduct of $k$-algebras is the free product, and the coequalizer of two maps $f, g : A \to B$ is given by quotienting $B$ by the two-sided ideal generated by elements of the form $f(a) - g(a)$.