The Dirac delta $\delta(x)$ is a continuous argument (generalized) function where $x$ is a continuous variable and can be time, space, frequency etc. Unfortunately, a simple definition like
$$\delta(x) =
\begin{cases}
\infty ~~~&,~~~ x=0 \\
0 ~~~&,~~~ x \neq 0
\end{cases}
$$
does or can not properly describe its behaviour. You need to define its properties under the integral sign, and that requires advanced mathematical stuff for a formal verification. But a simplified Riemann definition of $\delta(x)$ is given as:
$$\int_{-\infty}^{\infty}f(x)\delta(x-a) dx = f(a)$$ for a sufficiently smooth test function $f(x)$. All properties of $\delta(x)$ follows from this Riemann (seeming) integral operation.
On the other hand, the discrete counterpart, the unit-impulse or the Kronecker delta is a sequence of integer index $m$ with very well defined behaviour as:
$$\delta[m] = \begin{cases}
1 ~~~&,~~~ m=0 \\
0 ~~~&,~~~ m \neq 0
\end{cases}
$$
These two functions serve the same purpose for discrete and continuous systems, hence share the same name impulse. Yet they are two different, distinct entities; the Dirac impulse is a very problematic concept, whereas the discrete impulse is a mere simplicity.