Based on the definitions a Boolean domain denoted $\left\{0,1\right\}$ or $\mathbb B$ is a set with cardinality $2$ whose elements are interpreted as logical values,usually $0:=\text{false}$ and $1:=\text{true}$
Also a Boolean variable is a variable that takes its value from a Boolean domain.
It still is not clear to me what is a Boolean variable,so I decided to give myself an example:
The statement "There exist a real number $x$ satisfying the equation $x^2=-1$"is a false statement and the variable $x$ for which the statement is false should have the value $0$ in Boolean domain.
However I'm not sure if the example is right.
Wikipedia (current revision) gives a generalization of Boolean domain;
The Boolean domain $\left\{0, 1\right\}$ can be replaced by the unit interval $[0,1]$, in which case rather than only taking values $0$ or $1$, any value between and including $0$ and $1$ can be assumed. Algebraically, negation (NOT) is replaced with ${\displaystyle 1-x}$, conjunction (AND) is replaced with multiplication ${\displaystyle xy}$, and disjunction (OR) is defined via De Morgan's law to be ${\displaystyle 1-(1-x)(1-y)}$.
Can someone gives me some example of this generalization? How is it possible for variable to have a value between $0$ and $1$?
A Boolean variable is a variable that is either true or false. They take there values from the Boolean domain, often denoted as $\mathbb{B}=\{0,1\}$. Boolean variables follow the Boolean algebra which has the following basic operators:
\begin{array}{cccc} x&y&x\land y&x\lor y\\ \mathbf{0}&\mathbf{0}&0&0\\ \mathbf{0}&\mathbf{1}&0&1\\ \mathbf{1}&\mathbf{0}&0&1\\ \mathbf{1}&\mathbf{1}&1&1 \end{array}
\begin{array}{cc} x&\lnot x\\ \mathbf{0}&1\\ \mathbf{1}&0 \end{array}
In the above, $x\land y$ is interpreted as $x$ and $y$, while $x\lor y$ is interpreted as $x$ or $y$ or both. $\lnot x$ is interpreted as not $x$.
Boolean algebra can be used to describe logic circuits as well. In the following circuit, the top component is a NOT gate, the second to top an OR gate, and the bottom component an AND gate.
If the inputs $A$, $B$ and $C$ are Boolean variables, then the logic circuit outputs a logical $1$ precisely when $B$ and $C$ are true, or when $A$ is false. It can thus be expressed as $(0)$.
\begin{equation}\tag{0} Z=(B\land C)\lor\lnot A \end{equation}
Boolean variables are often used to express the truth of sentences in english, and statements about mathematics. For instance, "It is either raining outside or it is not", can be expressed with $(1)$, where $R$ is a Boolean variable expressing "It is raining outside".
\begin{equation}\tag{1} R\lor\lnot R \end{equation}
Once an English sentence has been expressed with mathematics, the vagueness is removed, and the truth of the sentence can be determined more easily.
In addition to assigning a Boolean variable to part of a sentence, entire sentences can be assigned to a Boolean variable. For instance we can define your example to be the Boolean variable $P$.
\begin{equation}\tag{2} P:=\text{There exist a real number $x$, satisfying the equation $x^2=-1$} \end{equation}
Since there is no real $x$ which satisfies $x^2=-1$, and $P$ is a Boolean variable which possesses a truth value, $P=0$.
The generalisation given on the Wikipedia page that you have linked is either missing information, or contains a mistake.
It states that the following:
However, consider the case where $x=0.5$: Here $\lnot x$ is $0.5$, and $x\land (\lnot x)$ is $0.25$; this is clearly a contradiction, since $x\land (\lnot x)$ is always false.