A finite limit is a limit over a finite diagram - that is, one whose shape is a finite category.
More generally, in higher category theory, a finite limit is a limit of a diagram that is a finite (n,r)-category.
A category that has all finite limits is called a finitely complete category or a (finitary) essentially algebraic theory.
A functor that preserves all finite limits is called left exact functor, a lex functor, a cartesian functor, or a finitely continuous functor.
The 2-category of finitely complete categories, left exact functors and natural transformations is often denoted Lex.
A product is a finite limit iff it is a finite product, hence if its factors are indexed by a finite set.
In particular, binary products are finite limits and the terminal object (being the limit over the empty diagram) is a finite limit.
Other common finite limits are pullbacks and equalizers.
The fixed locus of a group action is a finite limit if the group in question is a finite group.
The functor of geometric realization of simplicial sets into compactly generated topological spaces preserves finite limits (is a left exact functor), see there.
More generally, the functor of geometric realization of simplicial topological spaces (with respect to compactly generated topological spaces) preserves finite limits (by this Prop.).
For a category $\mathcal{C}$ the following are equivalent:
$\mathcal{C}$ has all finite limits.
$\mathcal{C}$ has all equalizers and binary products.
$\mathcal{C}$ has all pullbacks and a terminal object.
In fact, either class of limits may be expressed in terms of another, so that for a functor $F \;\colon\; \mathcal{C} \to \mathcal{D}$ the following are equivalent:
$F$ preserves finite limits.
$F$ preserves equalizers and binary products.
$F$ preserves pullbacks and the terminal object.
(The first statement may be found, e.g., in Borceux 1994, Prop. 2.8.2. From the proof there the second statement immediately follows.)
The equivalence of the first two items in Prop. is the finite analog of how a category with equalizers and all small products has all small limits.
(saturation to L-finite limits)
Prop. implies that finite limits are contained in the saturation of the class containing only finite products and equalizers, and also that of the class containing only pullbacks and terminal objects.
But the existence of finite limits in fact implies that of the larger class of L-finite limits, and this is the largest class of limits implied by the existence of finite limits (ParΓ© 1990, Prop 7, see there).
Therefore Prop. implies that L-finite limits constitute also the full saturation class of both equalizers+finite products as well as pullbacks+terminal objects.
Textbook account:
On the saturation to L-finite limits: