Econometrics is a branch of statistics that is applied to economics and financial economics. The key distinguishing feature of econometrics is that it deals specifically with time series data.
The problem with time series data is that it introduces many spurious correlations. If two data series both show a consistent trend over time they may appear correlated when they are not. They will may have a positive correlation coefficient although there is no causal link between them. This can also make it harder to find the true correlations.
Another problem with economic and financial data is that the data available may be limited, and it is not possible to perform experiments (as a physicist or a chemist would) to test hypotheses. Hypotheses usually have to be tested against existing historical data. This also makes the statistical techniques used to analyse the available data are more important.
Econometrics is hugely important to economics in general, and to financial economics in particular. It is largely the development of econometrics that has made economics a much more mathematical subject than it was a few decades ago.
Econometrics is used to both verify theories and to construct the models that are used to apply theories in many practical applications.