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As we know, a minimal projection must be Abelian, An Abelian projection must be finite. A minimal projection correspond to a rank one operator, a finite projection correspond to a finite rank operator, what kind of operator be corresponded with the Abelian projection?
I find some book use Abelian projections to define the type I von-Neumann algebras, but others use minimal projections instead. In general, how close are minimal projections to Abeilian projections?
You cannot define type I von Neumann algebras in terms of minimal projections, since many don't have them; $L^\infty[0,1]$, for example. What you probably saw is that some texts only care about factors, and in a factor the abelian projections are precisely the minimal ones.
Rank one projections (not "operators") are an example of a minimal projection. But minimality is a condition inside the algebra. If you consider an infinite projection $p$ and let $M=\{\alpha p+\beta(1-p):\ \alpha,\beta\in\mathbb C\}$, then $M$ is a von Neumann algebra with $p$ a minimal projection; I wouldn't call it a "rank one operator".
The same with finite projections. Finite rank projections are the finite projections of the von Neumann algebra $B(H)$. But all projections in a II$_1$-factor are finite, and no one would call those "finite-rank".
As $B(H)$ is a factor, its only abelian projections are the minimal ones. So there is no natural operators that you would call "abelian projections".
