As subrataray says, the Latin influence on English vocabulary has been 3-fold. Moreover, the Latin words were mixed up with the Greek in most cases, and hence it should be called the Latin and Greek influence.
The 1st phase of importing Latin words into English began long before the Angles and Jutes came to England, and the Germanic forefarthers of the English were still living in the Continent. The words that were imported during this phase were mainly related to plants, fruits, food; eg. wine, pea, plum, cheese, mint, street, cook, kitchen, milk. These did not necessarily exist in these forms, and the original words have undergone many changes.
The 2nd phase of Latin loan words came to England with the Christian Missionaries in the late 6th and 7th centuries A.D. They brought in with them ideas related to religion, the Church and monastic living. Eg: minster(L. monesterium), monk(L. monachus), bishop(L. episcopus), priest(L. presbyter), church(L.cyriacum). Most of these words were Greek, which entered English via Latin. During this phase the following words came from Latin to English :- apostle, disciple, Pope, archbishop, provost, abbot; shrine, cowl, pall, rule, mass, offer, altar, anthem, martyr, incarnate, pulpit, rosary, scripture.
With Christianity, clasical & scientific learning, concepts of law were also introduced. Eg: legal, prosecute, custody, zenith, index, mechanical, allegory, ornate, scripture.
The 3rd phase came with the great revival of classical scholarship in England in the 16th century. The words came over through the medium of writing, and were first absorbed and used by the learned, before filtering to the other classes.
1. Some words which had come from French were latinized:
verdit - verdict, perfet - perfect, doute - doubt, dette - debt, langage - language, avis - advice, aventure - adventure, al-amir(Arabic) - amiral(Fr & Mid E) - admiral, avril - april.
2. Many new words were invented on classical analogies.
a. native words + L. suffix : starvation, heathenism, talkative,etc
b. Gk word + L. suffix : climactic
c. L. prefix + native word : ex-king, interchange, preview, rebirth.
d. Eng proper names with latinized adjectives : Oxford - Oxonian, Shakespeare - Shakespearean.
3. Many Latin words and English synonyms which exist side by side
a. with same meaning : fire - conflagaration, ask - interrogate, truth - veracity.
b. have acquired slightly different connotations and are used in different contexts : same - identical, youthful - juvenile, readable - legible, manly - masculine, greatness - magnitude.
4. Some native nouns have English and/or Latin adjectives : father - fatherly, paternal; king - kingly, royal, regal; mouth - oral, nose -nasal, heaven - heavenly, celestial; water - watery, acquatic; mind - mental; moon - lunar; sun - solar; son/daughter - filial, eye - ocular; home - domestic.