I'm reading a book and it says that the $H_0^1(\Omega)$ space is defined as "the completion of $C_0^\infty(\Omega)$ w.r.t the Sobolev norm $\| \cdot \|_1$, where $C_0^\infty(\Omega)$ is the space of infinitely differentiable functions which are nonzero only on a compact subset of $\Omega$".
Can I simply understand the $H_0^1(\Omega)$ space as the space of all functions $u$ in $H^1(\Omega)$ whose value on the boundary of $\Omega$ is $0$ ($u\big |_{\partial\Omega}=0$)?
Intuitively you can, provided that $\partial \Omega$ is smooth enough.
In this book, page 273, theorem 2 exactly proves the statement you want. I don't know how much your background is of Sobolev space, but I suggest to read it. It is not hard. (Actually I would say it is really a good exercise to proof!)
Note that the functions in $H^1_0$ are defined a.e, so the "value at boundary" is not something immediatly defined. You can makes sense of a value at the boundary of function in $H^1$ with the trace operator if $\partial \Omega$ is smooth enough, but you need to do some work. The definition by the completion of $C^{\infty}_0$ gives you immediatly a well defined space.
