ANOVA is a procedure to compare nested linear models with normal errors. This is not restricted to the complexity of the model, so yes, the model can contain any number of (predictor/independent-) variables and consider any possible interaction between these variables. It could thus be even 4-way, 5-way etc. However, the interpretation of higher-order interactions (interactions between more than two variables) is (very) difficult.

"Local control" actually means that you organise the experiment to have "blocks" of measurements of factors that are influential but uninteresting and which vary and so add undesired noise to your measurements. This is easy to understand by example:

Consider you want to analyze the effect of a fertilizer on plant growth in the field. Plant growth depends on many things, particularily on soild conditions. Buth these will vary across the field. If you divide the filed in two halfs (one for control, one for the new fertilizer), you will get larger and smaller plants in each half due to differences in the local soil quality, so the data will be very noisy, making it difficult to recognize a systematic difference between the two groups. And even if you see a difference, it may be due to systematic differences in the soil quality between the two halfs of the field and not due to the fertilizer.

If you divide the large filed into several small "blocks", and in each block apply the fertilizer to one half of the block, the blocks will all be differtent due to the difference in soild quality. But within each (small) block, the soil quality is more homogeneous, what makes the comparison of the two groups within each block much more precise, as it eliminated much of the variation due to soill quality.

Such an experiment can be analyzed with an ANOVA that treats "treatment" (control/fertilized) and one factor plus "block" as another factor. So this is essentially a two-factorial ANOVA, what is also termed as a one-factorial ANOVA with a blocking factor (as only the treatment and not the blocking is interesting). The "block"-factor provides a "local control" (in this example you can take "local" literally!) of an influential factor (soil) that varies and statistically removes this variation (better: accounts for this variation) when analysing the treatment effect.

You can have a blocking factor for any number of intzeresting factors in your model, and you can even have several different blocking factors (imaging you have two different fields, or seedlings were obtained from different suppliers, or the data was measured over different years etc.).

A more recent way to allow "local control" is provided by mixed models that can model these factors as "random factors". This is much more flexible and powerful.

Thank you Sir for such a descriptive answer. It is really a great help. It is very useful information.

Also worthy of note is Jochen Wilhelm example that recalls the "ANOVA" origin to Fisher's studies (and agriculture).

It is true that the model can contain any number of (predictor/independent-) variables and consider any possible interaction between these variables; however, as Jochen Wilhelm said, the interpretation can be difficult if there are many variables.

In this case, for the young researcher Nitu Kaur , I would like to suggest the following link: https://www.itl.nist.gov/div898/handbook/pri/section3/pri334.htm