(Measure theory, functional analysis:)
Let (μ, X, B) be a measure space on a set X. As expected, the space is called finite if μ(X)<∞. (NOTE that the set X needn't be finite at all!) The space is called σ-finite (yes, that's a Greek letter participating in spelling an English word; mathematicians are weird that way...) if there exists a countable partition X=X1∪X2∪... (of X into measurable subsets Xn∈B) such that μ(Xn)<∞ for all n. That is, a measure space is σ-finite iff it is a countable union of finite measure spaces.
Often, just the measure μ will be called "σ-finite", when there can be no confusion regarding X and B.
For example, Lebesgue measure m on Rd is not a finite measure. However, m is σ-finite. You can see this e.g. by splitting Rd into countably many cubes Xn.
Many results of functional analysis and measure theory don't hold for just any measure space, but can be shown for finite measure spaces. But real analysts are much more interested in Rd, which is not finite; by extending the results to σ-finite spaces (something which is often rather easy to do, by utilizing the "continuity" of σ-additive measures) we can show them for the spaces of interest, such as Rd.