What does an error in ANOVA indicate? When I came across ANOVA, the instructor talked about df(Error), ss(Error), etc. What do these error terms indicate? Do error terms differ for two-way ANOVA with dependent and independent variables?

 A: Many models are based on a model for the dependent variable of the form "population mean + variation about the mean". Indeed, t-tests, one and two way ANOVA, multiple regression are all examples of this.
In the case of a two-way ANOVA with interaction, the model (in simplest terms) looks like this:
$$y_{ijk}=μ_{ij}+ε_{ijk}, $$
-- that is the $k$-th values at level $i$ of the "row" factor and level $j$ of the "column" factor (the IVs)  consist of a population mean for that combination of $i$ and $j$ and the individual variation about that mean (since the $k$th observation in factor-combination $i,j$  will not equal the population mean for that subgroup). 
Typically we decompose the mean for the two-way ANOVA into main effects and interaction: $μ_{ij}=μ+α_i+β_j+ (αβ)_{ij}$, giving:
$$y_{ijk}=μ+α_i+β_j+ (αβ)_{ij}+ε_{ijk}, $$
so that an observation consists of an overall (population) mean effect, plus a (population) "row" effect (representing deviations from that overall mean due to the row factor), a corresponding "column" effect, and interaction effect (an additional deviation for the particular factor-combination) and the individual variation from the mean.
Going back to the earlier form: $y_{ijk}=μ_{ij}+ε_{ijk},$ the individual variation about the population mean at factor-levels $i$ and $j$ is assumed to be a zero-mean, constant-variance random term, called the "error term".
It doesn't necessarily consist of actual errors in the ordinary sense of the word; the reasons for that are partly historical. It's just a description of the way the observations will vary from the population cell-means. That error term is an important part of the model. However, it may include things we would normally think of as error (such as measurement error in the DV). [The IVs are assumed to be measured without error, by the way, in the usual regression and ANOVA. This is usually not a problem for factors in ANOVA, especially where experiments are concerned.]
In normal theory inference (the usual confidence intervals and hypothesis tests), the error term is assumed to be normally distributed.

Now, why do we have $\text{SS(error)}$ and $\text{df(error)}$ and so on?
The variance of the $y$'s about the overall mean ($\mu$) is decomposed into portions explainable as variation of cell means about the population mean (variation of $\mu_{ij}$ about $\mu$) and random variation about the cell means (unexplained variability in the data). The first one is further decomposed into variance terms for row-effects, columns effects and interaction.
Now, if there really are no row, column or interaction effects at the population level, those variances for row, column and interaction will be non-zero due to the variation about the overall mean - they'll be relatively small, and the typical size is a function of the variance of the error term ($\text{var}(\varepsilon)=\sigma^2$) and we can even work out what distribution the estimates of these components of the y-variance should be. But if there are real row-, column-, and interaction- effects, those components of the y-variance will be typically larger and have a different distribution.
So to investigate the size of an effect (say the interaction-effect) in ANOVA, we compare the size of the implied value of $\sigma^2$ that would result if the effect was zero with the one from residual from the fitted model (the one that estimates $\text{var}(\epsilon)$ directly). The ratio of these two estimates of variance (the F statistic) will be (more or less) close to 1 if the effect is zero, and tends to be larger otherwise. 
We do the F-test to see if that ratio is bigger than could reasonably be explained by random variation (with no actual effect -- no interaction say). If it is, we'd reject the null hypothesis that the particular effect is zero.
This kind of calculation -- using ratios of estimates of variances to decide if effects that relate cell means are bigger than zero -- is called analysis of variance.
So terms like $\text{SS(error)}$ and $\text{df(error)}$ are central to figuring out whether there's evidence that the (IV) factors we're looking at really change the mean of the dependent variable or not.
A: I just wanted to add some information to @Glen_b's nice answer (+1). Perhaps, the OP knows that, but I would still clarify the terminology a little to the best of my knowledge/understanding.
$SS(error)$ represents error (residual) sum-of-squares and usually is referred to as $SSE$. Consequently, $df(error)$ represents degrees of freedom for error. I think that it is different from regression degrees of freedom. It is also my understanding that this term is, generally, different from degrees of freedom as a parameter for probability distributions. Moreover, it might be useful to note the existence of effective degrees of freedom (both regression and error/residual ones).
