model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
A model category is pointed if its underlying category is a pointed category, i.e., if the unique morphism from the initial object to the terminal object is an isomorphism, in which case both of them are denoted by $0$ (the zero object).
In any pointed category, one has a canonical zero morphism between any pair of objects $A$ and $B$, given by the composition $A\to 0\to B$.
The homotopy equalizer of $f\colon A\to B$ and $0\colon A\to B$ is known as the homotopy fiber of $f$.
The homotopy coequalizer of $f\colon A\to B$ and $0\colon A\to B$ is known as the homotopy cofiber of $f$.
In particuar there is the homotopy (co)-fiber of the zero object with itself, the loop space object- and reduced suspension-operation. Asking these operations to be equivalences in a suitable sense leads to the concept of linear model categories and stable model categories.
Model categories which are pointed without being linear or even stable:
(model categories of pointed objects)
Given any model category, its model category of pointed objects is a pointed model category.
In the case of the classical model structure on topological spaces this is the classical model structure on pointed topological spaces.
Pointed model categories which are stable:
Mark Hovey, Chapter 6 in: Model Categories, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books)
Section 4 – Homotopy fiber sequences in: Introduction to Homotopy Theory