TITLE: Motivation of filtered colimits
QUESTION [11 upvotes]: I am trying to move in categorical algebra beyond the basics. A Lawvere theory L is a small category with finite products. (I know that there also is a functor $(skeleton(FinSet))^{op}\to L$, which restricts a number of sorts in the algebraic theory to 1. Lets drop that requirement for now.) How to convert some variety to a Lawvere theory is pretty clear for me. The link (varieties ↦ Lawvere theories) is clear in some elementary operations, like

mapping an algebra by some functor F
↦ postcomposing F;
underlying functor
↦ precomposition of a functor between
Lawvere theories.

Then filtered colimits come. Lets take for reference “Adámek. a categorical introduction to general algebra.” Chapter 2 “Sifted and filtered colimits” and chapter 3 “Reflexive coequalizers” are devoid of mentioning varieties. Why the definition of a filtered colimit is such? I suppose there should be more concrete explanations involving algebraic operations, this is called “algebra” after all. Google suggests few texts on this subject, but they are abstract too. Any references?
The claim “an arbitrary algebra is a filtered colimit of finitely generated algebras” is needed to construct the left adjoint to an underlying functor. Can anyone refer me to its proof? (Update 2011-01-29. Also I want a precise proof constructing that left adjoint.) (Update 2011-01-29. Thank you all for insightful answers and comments. I suspect that there is no direct link between filtered colimits and traditional algebra, i.e. it is an abstract thing that is needed for another abstract thing… I need to think it through to formulate further questions.)

REPLY [5 votes]: The reason that Grothendieck originally considered filtered colimits and what is now known as the theory of accessible and locally presentable categories (I think named by Makkai and Paré) is as follows:
Let $x$ be an object of a category $C$ such that $hom_C(x,-)$ preserves $\alpha$-filtered colimits, then given any morphism $x\to colim F$ where $F:D\to C$ is an $\alpha$-filtered diagram, the morphism $x\to colim F$ factors through at least one $F(d)$ for some $d$ in $d$, and given any two factorizations through $F(d)$ and $F(d')$, there exists a majorant factorization through $F(d'')$ where $d''\geq d'$ and $d''\geq d$ extending the other two factorizations.  
I leave this as an exercise (it is, if you will, proof by introspection) (Hint: Use the corresponding statement for sets (which are valid because the statements hold for hom-sets) to perform the necessary manipulations).
This very powerful technique is used, for instance, in the modern generalizations of the small object argument and in situations regarding Bousfield localizations (the notion of accessibility is absolutely essential for results like Jeff Smith's theorem, for instance).  
See Clark Barwick's paper for a fairly detailed treatment with regard to its use in homotopy theory.  I would also suggest taking a look at Appendices 1 and 2 of Lurie's Higher topos theory as well as the book of Makkai-Paré, and also the standard modern reference on the subject by Adamek and Rosicky.