Edit: François asked me to include my comment within Gerhard's answer. I'll try to be brief. Just as algebraic theories can be described à la Lawvere as certain categories $T$ with finite products, and models of $T$ as finite product preserving functors $T \to Set$, it is of interest to consider "left exact theories" (aka "essentially algebraic theories") which are categories $T$ with finite limits, whose models are finitely continuous functors $T \to Set$. The theories of posets and of categories are examples of essentially algebraic theories.
It is known which categories are categories of models of some essentially algebraic theory, and this is the content of Gabriel-Ulmer duality. There is in fact a precise (bicategorical) contravariant equivalence between the bicategory of finitely complete categories, and the bicategory of locally finitely presentable categories. The concept of filtered colimit plays a crucial role in this development.
A sample theorem within this general theory which extrapolates Birkhoff's Variety Theorem is that a class of structures over a multi-sorted functional signature is a finitary quasi-variety (i.e., definable by Horn clauses over equational predicates) if and only if it is closed under products, subobjects, and filtered colimits within the category of all structures. For a reference, see Adámek and Rosicky, Locally Presentable and Accessible Categories, theorem 3.22.
Todd "I'm Happy to Oblige" Trimble