Variants
Variants
There are many different definitions of stratifications on spaces:

stratifications from [[filtered topological space filtrations]], 
[[posetstratified space posetstratifications]],  Whitney stratifications,
 ThomMather stratifications,

[[stratifold stratifolds]], 
[[orbifold orbifolds]],  homotopical stratifications (as developed by Quinn),
 conically smooth stratifications (as developed by AyalaFrancisTanaka,
 …etc.
Basic features
Stratification structures
Most of the above variants share the same basic structure, captured by the following definitions.
\begin{terminology} Given a decomposition $f$ of a [[topological space]] $X$ into [[subspace#topological_subspacessubspaces]] (denoted, in the following, by lowercase letters such as $s$), the exit path preorder $\mathsf{Exit}(f)$ of $f$ is the preorder of subspaces $s$ in $f$ with a generating arrow $s \to r$ whenever the closure of $r$ intersects $s$ nontrivially. \end{terminology}
\begin{defn} A stratification $f$ on $X$ is a decomposition of $X$ such that $\mathsf{Exit}(f)$ is a poset. \end{defn}
The subspaces in a stratification are also called strata. The opposite poset $\mathsf{Entr}(f)\op$ is also called the entrance path poset and denoted by $\mathsf{Entr}(f)$.
\begin{constr} Given a stratification $f$ on $X$, there is a map $X \to \mathrm{Exit}(f)$ mapping points $x \in s$ to $s \in \mathrm{Exit}(f)$, which is sometimes called the characteristic map of $f$, and (abusing notation) denoted by $f : X \to \mathrm{Exit}(f)$. \end{constr}
The characteristic map need not be continuous in general (unless the stratification is locally finite, see the next remark); it is, however, continuous on the preimage of any finite [[full functor  full]] subposet of the exit path poset. It is often convenient to construct stratificiations by constructing their characteristic map. 
\begin{eg} (Poset stratifications). Let $X$ be a space, $P$ a poset, and $f : X \to P$ a continuous map (in other words, $f$ is a $P$[[posetstratified spacestratification of $X$]]). This determines a stratification $\mathrm{c}(f)$ of $X$ whose strata are the connected components of the preimages $f\inv(x)$, $x \in P$. The map $f$ factors uniquely through the characteristic map $\mathrm{c}(f) : X \to \mathsf{Exit}(\mathrm{c}(f))$ by a [[conservative functorconservative]] poset map $\mathsf{Exit}(\mathrm{c}(f)) \to P$. (Such (characteristic,conservative)factorizations are essentially unique.) \end{eg}
\nid The example shows that any posetstratification determines a unique stratification. (The converse does not hold: many posetstratifications may determine the same stratification.)
\begin{eg} (Filtered spaces). Any filtered space $X_0 \subset X_1 \subset … \subset X_n$ in which $X_i$ is a closed subspace of $X_{i+1}$ defines a continuous map $X \to [n] = (0 \to 1 \to … \to n)$ mapping points in $X_{i+1} \setminus X_{i}$ to $i$, and thus a stratification by the previous example. As a concrete instance of this example, the filtration by skeleta of any [[cell complex]] defines the “stratification by cells” in this way. \end{eg}
\begin{eg}[Trivial stratification] Every topological space $U$ is trivially a stratification with a single stratum $U$. \end{eg}
\begin{rmk} (Continuity of characteristic map). Let $(X,f)$ be a stratified space. One says that the stratification $f$ is “locally finite” if each stratum $s$ of $f$ has an open neighborhood in $X$ which only contains finitely many strata. (If $(X,f)$ satisfies the frontier condition, see next remark, then, equivalently, $(X,f)$ is locally finite iff each point $x \in X$ has an open neighborhood intersecting only finitely many strata.) If $f$ is locally finite, then the characteristic map $f : X \to \mathsf{Exit}(f)$ is a [[continuous map]]. \end{rmk}
\begin{rmk} (Openness of characteristic map). Let $(X,f)$ be a stratified space. One says the stratification $f$ satisfies the “frontier condition” (or, as an adjective, that it is “frontierconstructible”) if, for any two strata $s, r$, the closure $\overline r$ intersects $s$ nontrivially, then $s \subset \overline r$. The stratification $f$ is frontierconstructible iff the characteristic map $f : X \to \mathsf{Exit}(f)$ is an [[open map]]. \end{rmk}
\nid It is generally very reasonable to assume stratifications to be locally finite and frontierconstructible.
The category of stratifications
\begin{defn} A stratified map $F : (X,f) \to (Y,g)$ of stratified spaces is a continuous map $F : X \to Y$ which factors through the characteristic maps $f$ and $g$ by a necessarily unique map, denoted by $\mathsf{Exit}(F) : \mathsf{Exit}(f) \to \mathsf{Exit}(g)$. \end{defn}
\nid Stratified spaces and their maps form the category $\mathbf{Strat}$ of stratification. The construction of exit path posets yields a functor $\mathsf{Exit} : \mathbf{Strat} \to \mathbf{Pos}$ (dually, using $^{\mathrm{op}} : \mathbf{Pos} \to \mathbf{Pos}). The functor has a right inverse, as follows.
\begin{constr} Every poset $P$ has a classifying stratification $P$ (also called the stratified realization of $P$), whose underlying space is the classifying space $P$ of $P$ (i.e. the [[geometric realizationrealization]] of the [[nerve]] of $P$), and whose characteristic map is the map $P \to P$ that maps $P^{\leq x} \setminus P^{<x}$ to $x$ (here, the full subposets $P^{\leq x} = {y \leq x}$ and $P^{<x} = {y < x}$ of $P$ are the “lower” resp. “strict lower closures” of an element $x$ in $P$). Moreover, given a poset map $F : P \to Q$, the realization of its nerve yields a stratified map $F : P \to Q$. We obtain a functor $ : \mathbf{Pos} \to \mathbf{Strat}$. \end{constr}
Every classifying stratification is frontierconstructible.
It makes sense to further terminologically distinguish maps of stratifications as follows.
\begin{rmk} (Types of stratified maps). Let $F : (X,f) \to (Y,g)$ be a stratified map. The stratified map $F$ is called: \begin{itemize} \item a substratification if $F : X \to Y$ is a subspace and $\mathsf{Exit}(F)$ is [[conservative functorconservative]]; if, moreover, $X = g\inv \circ \mathsf{Exit}(F) \circ f (X)$ then one says the substratification is constructible; \item a coarsening if $F : X \to Y$ is a homeomorphism (to emphasize the opposite process, one also calls $F$ a refinement); \item a stratified homeomorphism (or stratified iso) if $F : X \to Y$ is a homeomorphism of spaces and $\mathsf{Exit}(F)$ is an isomorphism of posets. \end{itemize} \end{defn}
Fundamental categories
Just as spaces have [[fundamental infinitygroupoid  fundamental]] $\infty$[[fundamental infinitygroupoid  groupoids]], stratified spaces also have “[[fundamental category  fundamental categories]]”. However, the role of sets for spaces is now played by posets: the following table illustrates the analogy. (The table is further explained below.) 
base concept  $\infty$concept  presentation  

[[set  sets]] $\equiv$ $(0,0)$category  \infty$sets $\eqv$ [[topological space  spaces]]  sets with w.e. 
posets $\equiv$ $(0,1)$categories  $\infty$posets $\eqv$ stratified spaces  posets with w.e.  
categories $\equiv$ $(1,1)$categories  $\infty$[[infinitycategory  categories]]  [[category with weak equivalences  categories with w.e.]] 
In the table, an “$\infty$X” is intuitively to be understood as an $(\infty,\infty)$[[(infinity,infinity)category  category]] which admits a [[conservative functor]] to an X, where X can e.g. stand for “set”, “poset”, or “category”. Yet more generally, X can be an $(n,r)$[[(n,r)category  category]] for $n,r < \infty$. A “set with weak equivalences” means a poset with weak equivalences in which each arrow is a weak equivalence. The left column is related to the middle column by an “$\infty$zation functor” (which simply includes $1$structures into $\infty$structures), and the middle and right columns are related by an “$\infty$[[category with weak equivalences#InfCatPres  localization functor]]” (which should be a weak equivalence in some sense). 
In order to make the above precise, one must work with sufficiiently convenient stratifications. We describe two simple ways of constructing/presenting fundamental $\infty$posets below.
Conical stratificiations
\begin{defn} Given stratifications $(X,f)$ and $(Y,g)$ their product is the stratification of $X \times Y$ with characteristic map $f \times g$. \end{defn}
\begin{defn} Given a stratification $(X,f)$, the stratified (open) cone $(\cone(X), \cone(f))$ stratifies the topological open cone $\cone(X) = X \times [0,1) / X \times {0}$ by the product $(X,f) \times (0,1)$ away from the cone point ${0}$ (here, the open interval $(0,1)$ is trivially stratified), and by setting the cone point ${0}$ to be its own stratum. \end{defn}
\begin{defn} A conical stratification $(X,f)$ is a stratification in which each point $x \in X$ has a neighborhood (i.e. an open substratification) that is a stratified product $U \times (\cone(Z),\cone(l))$ with $x \in U \times {0}$. \end{defn}
Every conical stratification is frontierconstructible.
\begin{constr} Given a conical stratification $(X,f)$, then [Lurie]{#LurieHA} constructs entrance path $\infty$category $\mathcal{E}\mathrm{ntr}(f)$ as a quasicategory: the $k$simplices of the quasicategory $\mathcal{E}\mathrm{ntr}(f)$ are precisely stratified maps $[k] \to (X,f)$, where $[k] = (0 \to 1 \to … \to k)$. \end{constr}
The construction translates objects in the middle row from the first column to the second column.
Regular stratifications
\begin{defn}[Regular stratifications] A stratification $(X,f)$ is regular if it admits a refinement $P \to (X,f)$ by the stratified realization of some poset $P$. \end{defn}
\begin{constr} Given a regular stratification $(X,f)$ and a refinement $P \to (X,f)$, one can construct the presented entrance path $\infty$category $\mathcal{PE}\mathrm{ntr}(f)$ as a poset with weak equivalences with underlying poset $P$ and weak equivalences $\mathsf{Exit}(F)\inv(\id)$. \end{constr}
(Showing that this construction is, in an appropriate sense, independent of the choice of $P$ requires a bit more work…)
The construction translates objects in the middle row from the first column to the third column in the above table.
References

{#quinn} F Quinn. Homotopically stratified sets. J. Amer. Math. Soc. 1 (1988), 441–499.

{#aft} David Ayala, John Francis, and Hiro Lee Tanaka. Local structures on stratified spaces. Advances in Mathematics 307 (2017): 9031028.

{#lurieHA} Jacob Lurie. Higher Algebra. 2012