Language families are groups of languages that are related to each other because they come from a common older language. The languages in a family have similar vocabulary or structure.
For example, French and Spanish both come from Latin. Latin was spoken a long time ago, and some of the people who spoke it started to speak different ways - for the word "good", one group started saying "bueno" while another group started saying "bon". Most words are said a little differently in Spanish and French, so the two are called different languages.
Most languages belong to a language family, but some languages do not. These languages are called language isolates.
There also are constructed languages, such as Esperanto. Constructed languages are made for different reasons (for example, making a 'world language', for fun, or for use in fiction, like with Klingon).