My impression is that this term is used loosely, without formal definition, to refer to the common situation where we have a forcing notion consisting of a partial order whose elements are all finite functions (that is, each on a finite domain) and the forcing order is functional extension. The most canonical example is Cohen forcing $\text{Add}(\omega,1)$ to add a Cohen real or actualy any number of Cohen reals. Other examples include the forcing to collapse a given cardinal to $\omega$ or the forcing to add a club set using finite conditions (not with countable conditions, which is another common way to do it). In some cases, there are examples where conditions are augmented with some other finite amount of information that still counts as forcing with finite conditions even though the order isn't literally functional extension.

Every forcing notion is equivalent, of course, to a forcing notion whose conditions are all finite, since we could replace any element $p$ in a partial order with $\{p\}$, which is a singleton set and hence finite, and then redefine the corresponding order on these singletons. So every condition in the new partial order is finite. But this literal interpretation is never what is meant by the phrase, "forcing with finite conditions," and I have never seen a formal definition of the concept going significantly beyond the loose understanding given in the examples above.

My impression is that this term is used loosely, without formal definition, to refer to the common situation where we have a forcing notion consisting of a partial order whose elements are all finite functions (that is, each on a finite domain) and the forcing order is functional extension. The most canonical example is Cohen forcing $\text{Add}(\omega,1)$ to add a Cohen real or actually any number of Cohen reals. Other examples include the forcing to collapse a given cardinal to $\omega$ or the forcing to add a club set using finite conditions (not with countable conditions, which is another common way to do it). In some cases, there are examples where conditions are augmented with some other finite amount of information that still counts as forcing with finite conditions even though the order isn't literally functional extension.

Every forcing notion is equivalent, of course, to a forcing notion whose conditions are all finite, since we could replace any element $p$ in a partial order with $\{p\}$, which is a singleton set and hence finite, and then redefine the corresponding order on these singletons. So every condition in the new partial order is finite. But this literal interpretation is never what is meant by the phrase, "forcing with finite conditions," and I have never seen a formal definition of the concept going significantly beyond the loose understanding given in the previous paragraph.