Just to clear up concepts, by visual inspection of the ACF or PACF you can choose (not estimate) a tentative ARMA model. Once a model is selected you can estimate the model by maximizing the likelihood function, minimizing the sum of squares or, in the case of the AR model, by means of the method of moments.
An ARMA model can be chosen upon inspection of the ACF and PACF. This approach
relies on the following facts: 1) the ACF of a stationary AR process of order p goes to zero at an exponential rate, while the PACF becomes zero after lag p.
2) For an MA process of order q the theoretical ACF and PACF exhibit the reverse behaviour (the ACF truncates after lag q and the PACF goes to zero relatively quickly).
It is usually clear to detect the order of an AR or MA model. However, with processes that include both an AR and MA part the lag at which they are truncated may be blurred because both the ACF and PACF will decay to zero.
One way to proceed is to fit first an AR or MA model (the one that seems more clear in the ACF and PACF) of low order. Then, if there is some further structure it will show up in the residuals, so the ACF and PACF of the residuals is checked to determine if additional AR or MA terms are necessary.
Usually you will have to try and diagnose more than one model. You can also compare them by looking at the AIC.
The ACF and PACF that you posted first suggested an ARMA(2,0,0)(0,0,1), that is, a regular AR(2) and a seasonal MA(1). The seasonal part of the model is determined similarly as the regular part but looking at the lags of seasonal order (e.g. 12, 24, 36,... in monthly data). If you are using R it is recommended to increase the default number of lags that are displayed,
acf(x, lag.max = 60).
The plot that you show now reveals suspicious negative correlation.
If this plot is based on the same that as the previous plot you may have taken too many differences. See also this post.
You can get further details, among other sources, here:
Chapter 3 in Time Series: Theory and Methods by Peter J. Brockwell and Richard A. Davis and here.