What linguists call "genetic relatives" are languages that were spoken natively by the same groups of people that later split up. Languages constantly evolve, and historically, new languages form when two dialects become so different that they are no longer mutually intelligible. These new languages are considered genetically related. Other types of language relations—word borrowing, pidgins, creoles, etc.—are the result of the influence of non-native speakers..
English is an Indo-European language, meaning that it comes originally from the Indo-Europeans, who split up over the ages into groups, which themselves split up, and eventually, settled in what is now England.
English is in the Germanic sub-family of Indo-European, meaning it is more closely related to other languages in the Germanic sub-family, such as German and Dutch, than it is to other Indo-European languages, such as the Romance languages, which include French, or the Indo-Aryan languages, such as Hindi and Urdu. English is related to Hindi and Urdu, though, as they are all Indo-European languages, but English is not related at all (by currently accepted linguistic theories) to non-Indo-European languages, such as Chinese, Turkish, and Hungarian. By "more closely related" what is meant in particular is that the population which first English split off from populations that speak other Germanic languages more recently than they split off from populations that speek non-Germanic Indo-European languages.
The language which has had the greatest non-genetic influence on English is, of course, French, due to the Norman conquest of England in the 11th century. Many words and phrases were borrowed from French at this time, but that is a non-genetic influence because French speakers at the time of the conquest had long ago split from the population of English speakers, and so relations between the language groups at that point are necessarily non-genetic.
The Wikipedia article about the history of the English language explains all this in much greater detail than I have explained it here.