Giants certainly do exist, however not much is known about them.
Giants occur in several of Tolkien’s works, but we never learn a great deal about them. Lúthien’s sleep-spell, already cited in reference to the beards of the dwarves (see p. 77), invokes ‘the neck of Gilim the giant’ and ‘the sword of Nan’ (BLT II.19) in its list of the longest things in the world, but little is known of either of these figures beyond the names.
The History of the Hobbit
In Tolkien's earlier works, he prominently features Giants, from Gilim mentioned above, to the giant who wanders Little Kingdom only to run into Farmer Giles. However, in his most famous work, Lord of the Rings (LR), Tolkien never had time for giants. Although the inclusion of creatures such as Trolls and Ents certainly suggests the possibility of the existence of "giants" in LR.
In early manuscripts, giants were certainly counted amongst the agents of Melkor:
At this point in the story the agents of Melko appear, the Úvanimor, ‘bred in the earth’ by him (Úvanimor, ‘who are monsters, giants, and ogres’, have been mentioned in an earlier tale, p. 75); and Túvo protected Men and Elves from them and from ‘evil fays’.
The Book of Lost Tales, Part 1
However, in the Parma Eldalamberon XIV it is mentioned that giants were later considered amongst Men.
Giants are clearly more prominent in early version of the Legendarium, however with the dissipation of the creatures, not much is known about them in the final, published works of Tolkien (at least during his life), their origins, however, are never made clear beyond them either being agents of Melkor or possibly men.
The stone giants from The Hobbit are discussed in this question.