Econometrics is the use of statistics on samples to derive the values of economic relationships (or any relationship between variables). Econometrics is a tool for the modern science of economics. In the days of: Keynes, Hayek, Marx, Pigou, Marshall, economics was essentially hand waving. People made claims about how the economy worked without using mathematical models - this meant theories could not be falsified. Modern day economics is about developing mathematical models which produce quantitative results - which can then be tested. The techniques of econometrics have been developed to test the results of models and deal with the problems that arise from economic data (namely that variables cannot be fixed).
A very important part of econometrics is hypothesis testing. So things like confidence intervals, significance levels are very relevant to metrics. Probability distributions are also very important as well - for both testing hypotheses and developing statistics.