Answer:
Language groups, or language families, are, as their name suggests, a way of categorising languages according to their origins. As there are several levels of categorisation, each language will belong to many different groups, for example:
French, Spanish, Portuguese, Romanian, Moldovan and Italian are Romance languages. Romance languages are in turn Italic languages, and Italic languages are members of the Indo-European family. So French is a Romance language, an Italic language and an Indo-European language.
English, in case you wanted to know, is also an Indo-European language (most languages based around Europe will be Indo-European), but it is West Germanic, meaning it is in the same family as languages such as German, Dutch and Swedish.
HERE ARE SOME OTHER INTERESTING LANGUAGE FAMILIES AND THEIR LANGUAGES
The Slavic languages, which is also Indo-European. They are Russian, Ukrainian, Bulgarian, Serbian, Croatian, Bosnian, Macedonian, Belarusian, Polish, Czech, Slovakian and Rusyn.
The Indo-Aryan languages are those spoken in India, Nepal, Bhutan and Bengladesh. The most spoken are Hindi, Tamil, Punjabi, Bengali and Nepalese. Pakistanis once spoke Indo-Aryan languages but no longer do, since they are in enmity with Indians and Bengalis.
The Semitic family, are the languages who's history goes all the way back to Biblical times and the times of the Jewish prophets. They are Hebrew and Arabic. For both languages were spoken as the official and active by Jewish prophets and they contain comse close words. Like "shalom" in Hebrew and "salam" in Arabic meaning "peace" or "salutations".
The Austronesian and Austro-Asian languages are those spoken in Southeast Asia and Ocania. The term "austro" means sailor, since people of Southeast Asia and Oceania were all skilled seafarers speaking similar dialects. They are Malay, Indonesian, Javanese, Filipino, Taiwanese, Vietmanese, Thai, Cambodian, Hawaiian, Samoan and Burmese. Dialects spoken in those countries are also in the Austronesian family of languages. Since Oceania and Southeast Asia are all islands and peninsulas, and their original inhabitans were all sailors and seafarers.
DID YOU KNOW?
That the Serbian, Croatian and Bosnian languages are basically all one language, known as "Serbo-Croatian?"
Some language families were once languages themselves. Like Romance, they were all once Latin.
Languages today were once dialects of their families. Spanish, Italian and Portuguese all started as dialects of Latin. Same with Russian and Ukrainian which were once simply dialects of the East Slavic language.
English is a Germanic language, but today it follows Latin-Romanic rules and even uses Spanish/Italian words for prefixes and suffixes? (Example, con means "with" in Spanish, its a prefix meaning "with" in English, "mal" is "bad" in Spanish, its an English prefix for "bad" and etc). English today has more cognates than its Germanic brothers.