Let $\mathcal{D}$ be a triangulated category. A t-structure on $\mathcal{D}$ is a pair of full subcategories
such that
for all $X \in \mathcal{D}_{\geq 0}$ and $Y \in \mathcal{D}_{\leq 0}$ the hom object is the zero object: $Hom_{\mathcal{D}}(X, Y[-1]) = 0$;
the subcategories are closed under suspension/desuspension: $\mathcal{D}_{\geq 0}[1] \subset \mathcal{D}_{\geq 0}$ and $\mathcal{D}_{\leq 0}[-1] \subset \mathcal{D}_{\leq 0}$.
For all objects $X \in \mathcal{D}$ there is a fiber sequence $Y \to X \to Z$ with $Y \in \mathcal{D}_{\geq 0}$ and $Z \in \mathcal{D}_{\leq 0}[-1]$.
Given a t-structure, its heart is the intersection
A t-structure on a stable (∞,1)-category is a t-structure on its homotopy category of an (∞,1)-category, according to def. 1.
(Higher Algebra, def. 1.2.1.4).
The heart of a stable $(\infty,1)$-category is an abelian category.
(BBD 82, Higher Algebra, remark 1.2.1.12)
If a the heart of a t-structure on a stable (∞,1)-category with sequential limits is an abelian category, then the spectral sequence of a filtered stable homotopy type converges (see there).
Related $n$Lab entries include Bridgeland stability?.
For triangulated categories
S. I. Gelfand, Yuri Manin, Methods of homological algebra, Nauka 1988, Springer 1998, 2003
Donu Arapura, Triangulated categories and $t$-structures (pdf)
Alexander Beilinson, Joseph Bernstein, Pierre Deligne, Faisceaux pervers, Asterisque 100, Volume 1, 1982