The third axiom of a model category reads (Homotopical Algebra, Dan Quillen https://www.springer.com/gp/book/9783540039143):
Fibrations are stable under composition, base change, and any isomorphism is a fibration.
What is a base change? There's a similar condition for cofibrations with something called a cobase change.
Base change is another name for the categorical pullback. The name stems from the fact that in algebraic geometry, you're often considering varieties/schemes $X$ over a fixed 'base' ring/field (or even another variety/scheme), say $k$. This intuitively means that the variety is defined by equations over $k$, and categorically, it means that $X$ comes equipped with a morphism $X\to\text{Spec}(k)$. Now, if you have an extension/map of 'bases' $k\to l$ for another ring/field $l$, you can 'change the base' from $k$ to $l$, which in the view of a variety as a bunch of polynomial equations just means you're interpreting equations over $k$ as equations over $l$. Categorically, though, it turns out to correspond to taking the pullback of the given $X\to\text{Spec}(k)$ along $\text{Spec}(l)\to\text{Spec}(k)$. Extrapolating from this example, sometimes a general pullback is called a based change.
Dually, a cobase change is the name for the categorical pushout.