What is difference between operating wave function with operator of an observable and measuring for an observable? People say operator of an observable helps in measuring for an observable. We also know that measuring leads to collapse of wave function. But operator on wave function gives a number times same wave function (which of course is not a collapsed wave function!). All intuitions I made about operator, wave function, measures, collapse are all seeming to be inconsistent. If operator doesn't collapse a wave function then what it is for. Is it just for calculating expectation value of observable. What in physical sense it is?
 A: There are a few different points/distinctions that need to be made here.
1. The expectancy value when measuring an observable to which the operator $A$ can be assigned is $\langle A \rangle = \langle \Phi | A | \Phi \rangle$ for a system described by a wavefunction $|\Phi\rangle$.
2. If the value $a$ is measured, the collapse of the wavefunction means that we project $ |\Phi\rangle$ onto the eigenspace of the eigenvalue $a$ of the operator, i.e., the wavefunction is changed.
3. $A | \Phi \rangle = a | \Phi \rangle$ for $a \in \mathbb{C}$ holds only if $| \Phi \rangle$ is an eigenfunction of the operator $A$. In general, $A | \Phi \rangle$ does not need to be proportional to $| \Phi \rangle$ but can be a different wavefunction.
4. An operator $A$ determines the possible values of its measurement variable by its eigenvalues and determines the possible wavefunctions after it has been measured by its eigenstates/eigenspaces.
5. Applying an operator to a wavefunction does not describe the measurement process.
A: You are right in pointing out that operating with an operator on a wavefunction gives a number times the wavefunction (assuming that it is indeed an eigenfunction of the said operator). Measuring the observable collapses the wavefunction. How a wavefunction collapses is still an open question in Quantum Physics. The job of the operator is to find out the possible eigenvalues of the wavefunction. This paper might help you with the progress that has been done in solving that open problem.   
