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When the Rings of Power were being handed out, the only races that were given magic rings were the Men, Elves and Dwarves.
It began with the forging of the Great Rings. Three were given to the Elves, immortal, wisest...fairest of all beings. Seven to the Dwarf Lords, great miners and craftsmen of the mountain halls. And Nine...nine rings were gifted to the race of Men who, above all else, desire power.
So why didn't Hobbits receive any Rings?
Hobbits (Perian) were only discovered around 2890 years after the Rings had been forged. Therefore Sauron was unaware of their existence and wouldn't've been able to give them any rings.
The Rings were forged between 1500 and 1600 of the Second Age.
1500 The Elven-smiths instructed by Sauron reach the height of their skill.
They begin the forging of the Rings of Power.
1590 The Three Rings are completed in Eregion.
1600 Sauron forges the One Ring in Orodruin. He completes the Barad-dûr.
Celebrimbor perceives the designs of Sauron.
Return of the King: Appendix B, Tale of Years
The Hobbits were only discovered early in the eleventh century of the Third Age.
1050 Hyarmendacil conquers the Harad. Gondor reaches the height of its
power.
About this time a shadow falls on Greenwood, and men begin to call it Mirkwood.
The Periannath are first mentioned in records, with the coming of the Harfoots to
Eriador.
ibid.
In short, they were originally intended for Elven leaders, but eventually they were given to the rulers and kings of Men and Dwarves. Hobbits had no rulers, and they were barely noticed most of the times.
There are a total of 20 rings, 4 of which forged in secret. The first 16, were some sort of practice, and the Elven smiths, Celebrimbor, and Sauron worked on them. Sauron used these opportunity to bind them to his own ring of power secretly, which he forged in Mount Doom. These 16 rings were in Elven hands, and meant to be worn by Elves. Then, Celebrimbor alone created 3 Elven rings of power, not bound to the One Ring.
When Sauron put on his ring of power, the Elves sensed it, and removed theirs. Sauron was mad, for his plan had failed, and entered war on the Elves, demanding the return of the rings. The Elves managed to hide the 3 most powerful rings, but Sauron eventually retrieved the other 16.
He decided to give them now to Men, and Dwarves, who he believed would bow to his will more easily. The 9 kings, at first drew great power from the ring, but eventually became Nazgûl. The Dwarves, at first achieved great riches, but the rings just made them grow more greedy, and that lead them to their ruin.
The 3 Elven rings were passed on. Gandalf, Galadriel, and Elrond were the bearers of them by the time of LotR.
What was Sauron's original plan? It's hard to tell. We can't say if he intended to create more rings of power after using the original 16 for controlling the Elves. We can't tell if he intended to have more made for Men and Dwarves, or if he simply thought Elves would suffice. His plan fell through though, and he had to improvise. Hobbits didn't have a great weight in Middle-earth though. There were no leaders to control or target.
Sauron gave rings to the three races of beings that were given life at the start of The World -- Elves and Men were created by Ilúvatar, and Dwarves were created by Aulë, but given life by Ilúvatar. Thus, these three are the "eldest" races of beings on Earth (not counting the Ainur, who were created before the world).
The other races -- Ents, Orcs, Hobbits, etc -- were all either created later by other Ainur, or evolved naturally from one of those three. Hobbits, in particular, were merely an offshoot of Man that diverged some time in their distant past.
Even the Hobbits themselves don't actually know when or how they came to be a separate people from humans, but the earliest known record of their existence (that I know of) came in the Third Age, long after the rings had been created.
So, the most likely reason the Hobbits didn't get rings is because, at the time, Hobbits were still just a slightly shorter group of humans.
Echoing Edlothiad's answer, the simple fact is that Sauron was most likely unaware of their existence. Gandalf certainly thinks so, and the roundabout way that his Nazgûl go about locating Bilbo would strongly imply that The Enemy was unable to give them any useful intelligence about the likely location of the Hobbit-folk
'Yes, alas! through him the Enemy has learned that the One has been found again. He knows where Isildur fell. He knows where Gollum found his ring. He knows that it is a Great Ring, for it gave long life. He knows that it is not one of the Three, for they have never been lost, and they endure no evil. He knows that it is not one of the Seven, or the Nine, for they are accounted for. He knows that it is the One. And he has at last heard, I think, of hobbits and the Shire. 'The Shire - he may be seeking for it now, if he has not already found out where it lies. Indeed, Frodo, I fear that he may even think that the long-unnoticed name of Baggins has become important.'
The Fellowship of the Ring: Chapter 1 - A Long-expected Party
Prior to this there simply aren't any records of the Hobbits. They simply seem to be rural folk living along the rivers and in the moorland in the vicinity of Bree.
For it was in the one thousand six hundred and first year of the Third Age that the Fallohide brothers, Marcho and Blanco, set out from Bree; and having obtained permission from the high king at Fornost, they crossed the brown river Baranduin with a great following of Hobbits. They passed over the Bridge of Stonebows, that had been built in the days of the power of the North Kingdom, and they took all the land beyond to dwell in, between the river and the Far Downs. All that was demanded of them was that they should keep the Great Bridge in repair, and all other bridges and roads, speed the king's messengers, and acknowledge his lordship.
The Fellowship of the Ring: Prologue
With no obvious leadership (aside from family heads-of-household like Gollum's 'grandmother') and no regional influence, there's little point corrupting one of them with a ring of power.
There was among them a family of high repute, for it was large and wealthier than most, and it was ruled by a grandmother of the folk, stern and wise in old lore, such as they had.
The Fellowship of the Ring: Chapter 1 - A Long-expected Party
The Hobbits are a small people (in both stature and number) living far from Mordor in places that are largely devoid of any strategic use, behind an Elf-forest, behind the Misty Mountains and along river systems that don't connect with Mordor. In short, they were overlooked.
