I am reading an article about solving large sparse linear systems, in this paper it’s said that most of the iterative methods to solve $Ax = b$ are very much influenced by the spectral properties of matrix $A$. Is the condition number one of spectral properties? What else might influence directly on solving $Ax = b$ via iterative methods that is a spectral properties of $A$? 



Is spectral properties a general term for condition number?
"Spectral properties" in general means "anything which is related to the eigenvalues (spectrum) of the matrix". This includes the condition number, but in general means much more.


The convergence of iterative methods is influenced deeply by the location of the eigenvalues of the system matrix, not only through the condition number.


For a normal matrix, the GMRES residual at step $i$ satisfies
$$
||r\_i||^2 = \min\_{p\in P\_i} \sum\_{j=1}^n |p(\lambda\_j) \rho\_j|^2,
$$
where $\rho\_j$ are the components of $r\_0$ in the eigenvector basis, and $P\_i$ is the set of polynomials of degree $\leq i$ normalized with $p(0)=1$. So if the eigenvalues are "easily approximated" with a low-degree polynomial, then convergence is fast. For instance, if the eigenvectors are clustered around $m$ points in the complex plane, then there is a degree-$m$ polynomial in $P\_m$ which vanishes on these $m$ points: so we expect $p(\lambda\_j)$ to be small for all $j$, and the residual to decay sharply after $m$ iterations.


The convergence behavior of iterative methods depends on how well this polynomial approximation problem can be solved. This, in turn, depends on the location of each eigenvalue, not only on the condition number: clustered eigenvalues work better than equispaced ones, for instance.


The theorems that relate convergence to the condition number (such as the $\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}$ estimate for CG) are only rough bounds which are often far from the real residuals. 


For non-normal matrices, the situation is more complicated. Experimentally, spectral properties are often related to convergence in a similar way (for instance, to assess the effectiveness of a preconditioner $P$ it is common to check how far the eigenvalues of $P^{-1}A$ are from 1), but there are edge cases in which this does not work. Greeenbaum and Strakos proved that there exist matrices with any prescribed set of nonzero eigenvalues, and any prescribed (nonincreasing) succession of residual norms $||r\_i||$, so a result of that kind is impossible to obtain. It is difficult to give blanket results on the convergence of GMRES for nonnormal matrices. What you will find in most papers is numerical results which prove that on a suitable sequence of problems the number of needed iterations is constant irrespective of their the dimension (or grows slowly).


