Katzarkov-Kontsevich-Pantev define a smooth dg ($\mathbb{C}$-)algebra $A$ to be a dg algebra which is a perfect $A \otimes A^{op}$-module. They say that an $A$-module $M$ is perfect if the functor $Hom(M,-)$ from ($A$-modules) to ($\mathbb{C}$-modules) preserves small homotopy colimits. (Definition 2.23 of KKP) *
Kontsevich-Soibelman define smooth dg (or $A_\infty$) algebras in the same way; but they define perfect $A$-modules to be ones which are quasi-isomorphic to a direct summand of an extension of a sequence of modules each of which is quasi-isomorphic to a shift of $A$. (Definition 8.1.1 of KS) **
First question: I have trouble wrapping my head around what it means for a functor to preserve small homotopy colimits. I don't understand the definition from Kontsevich-Soibelman, either. What are the "moral" meanings of these conditions?
Second question: Presumably these two conditions are equivalent. How do you prove this?
Third question: If $A$ is a commutative $(\mathbb{C}$-)algebra, then presumably smoothness of $A$ as a dg algebra is equivalent to smoothness of $\operatorname{Spec} A$ as a ($\mathbb{C}$-)scheme. (Maybe you need $A$ to be finite type?) How do you prove this? Furthermore, Example 8.1.4 of KS gives some more examples of dg algebras that are allegedly smooth (for instance free algebras $k\langle x_1, \dots, x_n \rangle$). Again, I don't know how to prove that these are in fact smooth, and Kontsevich-Soibelman don't seem to provide proofs; or maybe I overlooked something.
*The answers below indicate that this is the standard definition of compact (also known as small) module.
**The answers below indicate that this is the standard definition of perfect module.