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.