Martingale is a stochastic process with the martingale property. If we have \(X_t\) as the stochastic process, the martingale property says that \(\mathbf{E}[X_t\mid\mathcal{F}_s] = X_s\), for \(s\lt t\). Closely related to this is the local martingale. However, the Wikipedia page does not have it clearly explained. Here is my narrative.

A martingale is a stochastic process that (1) adapted to the filtration
\(\mathbb{F}\), (2) uniformly integrable, and (3) exhibits the martingale
property. The filtration part is easy to understand, simply the information up
to time \(t\) is enough to tell about the stochastic process \(X_t\). The
uniformly integrable property simply means there is only a finite amount of
mass to go around.
Technically, it means
\(\forall \epsilon>0,\ \exists K: \mathbf{E}[\lvert X_t\rvert \cdot \mathbf{1}\{\lvert X_t\rvert \ge K \}] < \epsilon\).
Note that since \(X_t\) is a stochastic process, the \(\epsilon\) and \(K\)
above applies to all time index \(t\). This may not hold for a heavy-tail
distribution. If the martingale \(X=\{X_t: 0\le t<\infty\}\) is uniformly
integrable, we can have the *closure property* such that \(X_\infty\) can be
defined and append to \(X\) and we can reverse to compute
\(X_t=\mathbf{E}[X_\infty\mid\mathcal{F}_t]\).

A localized stochastic process is defined as follows. If the set \(C\) is a family of stochastic processes of certain properties, the localized set \(C'\) is a superset of \(C\) such that, if \(X_t\in C'\) then we can find an increasing sequence of stopping time \(\tau_n\) such that \(\lim_{n\to\infty}\tau_n=\infty\) and the stopped processes \(X_{\min(t,\tau_n)}\in C\). This is an example of a localized stochastic process: If \(C\) is the set of all bounded stochastic processes, then the Brownian motion \(W_t\notin C\) as it is not bounded. However, we can define the stopping time \(\tau_n=\inf\{t: \lvert W_t\rvert = n\}\) such that the stopped process \(W_{\min(t,\tau_n)}\in C\). Therefore \(W_t\in C'\) is a locally bounded process.

A local martingale is simply a stochastic process which when localized is a martingale. The Wikipedia page use the following as an example:

\[X_t = \begin{cases} W^T_{t/(1-t)} & 0\le t < 1, \\ -1 & 1\le t < \infty. \end{cases}\]The stopping time \(T=\inf\{t: W_t=-1\}\) is the first hitting time of \(W_t=-1\). The stopped process \(W^T_{t/(1-t)}\) is a “compressed” Brownian motion started at \(W_0=0\) until it hits \(-1\) the first time than holds this value constant. Since a Brownian motion \(W_t\) will hit \(-1\) at some time \(t\) almost surely, we can have \(W^T_\infty=-1\) a.s., and hence the “compressed” version \(W^T_{t/(1-t)}=-1\) before \(t=1\). Thus the limit agrees on both side:

\[\lim_{t\to 1} W^T_{t/(1-t)} = -1.\]Brownian motion \(W_t\) exhibits martingale property, i.e. \(\mathbf{E}[W_t\mid\mathcal{F}_s]=W_s\), so as the stopped process \(W^T_t\). But this does not apply to the example process \(X_t\) because at least this expectation does not hold at \(t=1\) and \(s=0\), which \(X_0=W^T_0=0\) and \(X_1=-1\) so \(\mathbf{E}[X_1\mid\mathcal{F}_0]\ne X_0\). However, it is a local martingale when we consider the sequence of stopping times

\[T_n = \frac{n}{n+1} \mathbf{1}\{T \ge n\} + (\frac{T}{T+1} + n)\mathbf{1}\{T<n\}.\]In summary, a local martingale is a relaxed version of martingale which exhibits the martingale property often except in some corner cases.