Econometrics is the application of statistical methods to economic data and whose purpose is to apply, test or refine economic theories. This data may concern macroeconomic aggregates (gross domestic product, household consumption, business investment, employment, etc.), financial markets (closing prices, trading volumes, etc.), or microeconomic datasets (individual income, emoployment status, marital status, academic background, etc.).
A very large part of econometrics focus on regression analysis. If you ever attended an introductory lecture in economics, you have been exposed to supply and demand models. Those models treat pairs of traded quantities and prices as equilibrium values and assigns influence to various factors on prices and quantities through their respective impact on the behavior of those who demand a good or service and on the behavior of those who supply it. This sort of model inherently comes with a problem: demand and supply jointly determine prices and quantities, which means that if you want to study either the demand or the supply in isolation you have to untangle them. The trick here is that if you could find something that would presumably make the supply, but not the demand move, you can use its variations to "trace out" a demand schedule. For example, too many freezing days in Florida will have something to do with the supply of orange products, but it's unlikely that it will impact simulatenously the demand of those products -- especially if you look at a market for them outside of Florida.
In a regression framework, regressing quantities on prices, prices happen to be an endogenous regressors -- by the very design of a supply and demand model, they are determined jointly with quantities. For the orange juice market example, freezing days in Florida would be an instrumental variable... In economics, we have a lot of models that involve simultaneity and endogeneity problems like these which is why you will see loads of papers in econometrics dealing with regression analysis with an eye to specifically solve those problems.
So, in general, you could say that econometrics is a special kind of statistics, one that seeks to solve specific types of problems. I only talked about one kind of problem that can emerge in both cross-sectional and time-series data, but there are obviously many more problems that emerge because of the sort of research we do in economics. A lot of work in regression analysis has been done by econometricians because we needed the tools to help us analyze economic data.