By the time of the events in The Fellowship of the Ring, Hobbits have lost track of where they came from originally, as a people.
They're clearly closer to humankind than they are to Elves or Dwarves, though.
There are three kinds of Hobbits: the Harfoots, who live near the mountains and enjoy holes in the ground and the company of Dwarves; the Stoors, who like riverbanks and are "less shy of Men" (prologue.1.10); and the Fallohides, who are a bit taller than other Hobbits and hang out with Elves.
The Harfoots are the most common kind of Hobbit: they still live underground, and they like to settle in one place. Close to humankind, eh? We'll see about that.
Hobbits count time starting with the year they crossed the River Brandywine (a.k.a. Baranduin) into the Shire.
The Shire is a large square of land which is also the center of Hobbit life and culture in the Third Age (the age in which the events of The Lord of the Rings take place).
Relative to the other lands of Lord of the Rings (Gondor, Rohan, Mordor, etc.), the Shire is in the northwest, in the region of Eriador. (Check out a map to figure out where all of these places are.)
The Shire has been peaceful for centuries; in fact, the narrator assures us that "At no time had Hobbits of any kind been warlike, and they had never fought among themselves" (prologue.1.17). Impressive.
Underneath the Shire are a ton of Hobbit holes (called smials), many of which house huge Hobbit families (like the Tooks or the Brandybucks). Pretty cool digs.