Physical interpretation of Uncertainty Uncertainty of an operator $\hat{A}$ when observing a state $|\psi\rangle$ is defined as
\begin{equation}
\Delta A_{\psi} = |(\hat{A} - \langle\hat{A}\rangle)\psi|
\end{equation}
Now assume that there is some uncertainty for an operator $\hat{A}$ i.e., when I operate it on a state $|\psi\rangle$ there will be some error every time I measure it.(This is what I think physical interpretation of having uncertainty is)
How does this statement and the definition of uncertainty imply each other. i.e., how is this statement and the definition of uncertainty same?
 A: 
when I operate it on a state $|ψ⟩$ there will be some error every time I measure it

This is a common misconception. Performing a measurement on a system is not mathematically represented by applying the corresponding Hermitian operator to that state. The operator just tells us the possible measurement outcomes through its eigenvalues, and it allows us to determine the probability of measuring each of those values by expressing our state in the eigenbasis of the operator. $\hat A|\psi\rangle$ will not give you the measurement outcome.

How does this statement and the definition of uncertainty imply each other. i.e., how is this statement and the definition of uncertainty same?

Uncertainty and error are two different things in the context of Quantum Mechanics. For a given state we can compute the uncertainty of a measurement as you have described. What this means is that if we were to prepare a bunch of similar states and measure the observable in question, we would find the standard deviation of those measurements to (ideally) have a value of $\Delta A_\psi$. Error would come into play in the method of measurement, where the returned values have some error associated with them. Uncertainty is "baked into" Quantum Mechanics and is independent of the method of measurement, whereas error depends on how the measurement was done. A system can have no uncertainty with respect to an observable yet the measurements will still have non-zero error.

To address another confusion, uncertainty is not the same thing as the uncertainty principle, which relates the uncertainties of two observables. For a given state, we can determine the uncertainty of a single observable. Or, using the language from above, we can determine the standard deviation of measurements of an observable from similarly prepared states.
A: There is always uncertainty associated with measurements,a nd which is analyzed using statistical methods. This uncertainty however can be made (in principle) vanishingly small with improving equipment. I say in principle, because there are obviously some limitations - e.g., our measurement cannot last longer than the lifetime of the universe, etc.
The uncertainty that one deals in quantum mechanics is however of different nature, resulting from the laws of nature rather than the imperfection of our measuring apparatus. Thus, this uncertainty can never be eliminated by simply improving our measurement device. Yet, when carrying out actual measurements it is calculated in the same way - in this respect one has to distinguish the mean and the sample average. See this answer for some additional details.
A: No, you should not think of the uncertainty principle as meaning that measurements necessarily have associated errors- it is more fundamental than that. What the uncertainty principle means is that there are certain combinations of properties- such as position and momentum- for which a particle can never have simultaneous exact values. It is not that the particle has exact values but we cannot measure them- it is that the particle does not actually have exact values.
