Where English sits on the family tree
Glottolog, an open catalogue of the world's languages, gives English this lineage: Indo-European, then Classical Indo-European, Germanic, Northwest Germanic, West Germanic, North Sea Germanic and Anglo-Frisian (Glottolog: English). German and Dutch share the path down to West Germanic and then split off. Western Frisian goes one step further with English, into the Anglo-Frisian group.
Romance is on a different limb. Glottolog files it under Indo-European, then Classical Indo-European, Italic, Latino-Faliscan, Latinic and Imperial Latin (Glottolog: Romance). French, Spanish, Italian, Portuguese and Romanian all appear inside that Romance node. The two paths share only their first two steps, so English and the Romance languages meet at Classical Indo-European and part ways immediately below it, one going into Germanic and the other into Italic. Russian and Hindi also sit under Classical Indo-European, in other branches, which makes French or Spanish no closer kin to English, by this tree, than those languages are.
So the honest one-line answer is: English and French are cousins many times removed, while English and Frisian, German or Dutch are close relatives.
What makes a language Germanic or Romance
Family membership is a statement about descent. A Romance language is one that grew, generation by generation, out of the spoken Latin of the Roman Empire. A Germanic language is one that grew out of the common Germanic speech of northern Europe. Borrowed words do not change that line of descent, in the same way a family that buys Italian furniture is not suddenly Italian.
Cambridge University Press's A History of the English Language (edited by Richard Hogg and David Denison) describes English as "a dialect of the Germanic branch of Indo-European" and says the Germanic-speaking Anglo-Saxons brought the language to Britain as immigrants (Hogg and Denison, A History of the English Language, chapter 1 excerpt). The same chapter concludes from both the family tree and the wave model of language relationships that the closest language to English in purely linguistic terms is Frisian, spoken by about 400,000 Frisian-Dutch bilinguals in the Dutch province of Friesland and a few thousand people in Germany.
That last point is a useful test. If English were a Romance language, its nearest neighbor would be French or Spanish. Instead it is a small language of the North Sea coast that most English speakers have never heard.
The sound evidence: father, three, hundred
Linguists do not rely on impressions of which words look familiar. They compare regular sound correspondences, and the Cambridge chapter walks through a classic example. Lined up side by side, the words for father, three, hundred and who show a pattern across Sanskrit, Greek, Latin and Old English.
Where Sanskrit, Greek and Latin have /p/, English has /f/: Latin pater, Old English fæder, modern father. Where they have /t/, English has the th sound: Latin trēs, Old English þrēo, modern three. Where Greek and Latin have /k/, English has /h/: Latin centum, Old English hund, modern hundred. The chapter notes this shift is called Grimm's Law, after the nineteenth-century linguist Jacob Grimm.
This is the fingerprint of Germanic. A word that entered English from Latin or French later, after the shift had happened, keeps its original consonant. That is why English has both father (inherited) and paternal (borrowed), and both hundred and century. The pairs show two layers in the language: an inherited Germanic one and a borrowed Romance and Latin one sitting on top.
How far back the record goes
The classification does not rest on guesswork about prehistory alone. The Cambridge chapter points out that the oldest English texts, apart from tiny fragments, date from about AD 700, and that the only older Germanic texts are in Gothic, about 200 to 300 years earlier. Its comparison table lists Old English forms such as fæder and þrēo, and those forms already show the Germanic consonants.
Comparative reconstruction then fills in what came before writing. The chapter explains that the method works most clearly in sounds, word forms and vocabulary, and that even the syntax of Germanic and of Indo-European has been reconstructed in some detail. Its authors write that much of what we think we know about the history of English is held in place by a mesh of interlocking hypotheses so tight that its correctness is virtually certain, while adding that there is still a great deal left to discover.
Why so many English words look Latin or French
The borrowed layer is large. The Cambridge chapter reports, from searches of the second edition of the Oxford English Dictionary, about 24,940 words sourced from Latin and 9,470 from French, compared with 1,530 from Scandinavian languages and 1,280 from Spanish. The authors warn that such figures remain imprecise and that exact counts are unattainable.
The same chapter adds a caution that matters for this question. In a note on that OED table, the authors say the already-existing English language did not get its basic vocabulary and structure from any of the languages listed, Latin and French included, and that the table is there only to show the variety of sources English has drawn on. Note also the ratio inside the table: Latin accounts for well over twice as many words as French.
Historically, the chapter links much of this to the Norman Conquest, which brought significant numbers of French-speaking settlers to England, after earlier Scandinavian settlements in the eighth and ninth centuries.
A quick test with real etymologies
You can see the two layers by checking etymologies in Wiktionary, which traces each word back step by step. Water comes from Middle English water, from Old English wæter, from Proto-West Germanic watar. Drink goes back to Old English drincan and Proto-Germanic drinkaną. House is likewise inherited from Middle English hous. None of these passed through Latin or French.
Compare government: Middle English governement, from Old French governement, and Wiktionary notes it displaced several native Old English words. Education was borrowed from Middle French éducation, from Latin ēducātiō, "a breeding, bringing up, rearing." Paternal came from Old French and Latin paternus, from pater, and century from Old French centurie and Latin centuria, from centum, one hundred.
Put those next to the Grimm's Law list and the pattern is plain. Father and paternal, hundred and century: in each pair the first word is the inherited Germanic form and the second a later loan built on the Latin root. A Romance language would not have that split, because its inherited words are themselves the Latin-derived ones.
If you are learning a Romance language as an English speaker, the borrowed layer is a real advantage, since formal words like education and government have obvious counterparts in French. Grammar is another matter. See our guides to French articles and ser vs estar for two places where Romance grammar works differently from English.