Here are the relevant passages:
Miss Smith, and Miss Bickerton, another parlour boarder at Mrs. Goddard's, who had been also at the ball, had walked out together, and taken a road, the Richmond road, which, though apparently public enough for safety, had led them into alarm.—About half a mile beyond Highbury, making a sudden turn, and deeply shaded by elms on each side, it became for a considerable stretch very retired; and when the young ladies had advanced some way into it,they had suddenly perceived at a small distance before them, on a broader patch of greensward by the side, a party of gipsies. A child on the watch, came towards them to beg; and Miss Bickerton, excessively frightened, gave a great scream, and calling on Harriet to follow her, ran up a steep bank, cleared a slight hedge at the top, and made the best of her way by a short cutback to Highbury. But poor Harriet could not follow. She had suffered very much from cramp after dancing, and her first attempt to mount the bank brought on such a return of it as made her absolutely powerless— and in this state, and exceedingly terrified, she had been obliged to remain
How the trampers might have behaved, had the young ladies been more courageous, must be doubtful; but such an invitation for attack could not be resisted; and Harriet was soon assailed by half a dozen children, headed by a stout woman and a great boy, all clamorous, and impertinent in look, though not absolutely in word.—More and more frightened, she immediately promised them money, and taking out her purse, gave them a shilling, and begged them not to want more, or to use her ill.—She was then able to walk, though but slowly, and was moving away—but her terror and her purse were too tempting, and she was followed, or rather surrounded, by the whole gang, demanding more.
After this, Harriet is rescued by Frank Churchill, whereupon Emma tells her that she will let Mr. Knightly know that "such people" have taken up residence in the area; Mr. Knightly will then presumably take steps to deal with them.
Some more detailed questions:
Austen was likely referring to Romani, and she and her readers would have been fairly familiar with them as a concept despite being unlikely to have ever actually spoken to a Romani person beyond perhaps a quick transaction. In some ways, they were in the same position as Jews: "foreign" Others who were in reality British citizens, but with their own customs and cultures, frequently used in literature but rarely allowed to be well-rounded characters.
The Romani arrived in Britain at the beginning of the sixteenth century, probably after they were expelled from France in 1504. The earliest references, generally account books from aristocratic estates where they entertained with music, dancing, and fortune-telling for pay, call them "Egyptians" - their assumed nationality of origin. After a few decades, however, antiziganism was in full force, and England's Romani population found themselves a persecuted minority associated with crime and begging. Henry VIII attempted to expel them in 1530, but his measures had little effect.
Despite scholars developing a strong theory of Romani descent from India, the idea of their origins being shrouded in mystery stuck hard in the popular consciousness. Plots in literature that involved them often reflected this thematically: Sir Walter Scott's Guy Mannering features a young man who was raised by smugglers without knowledge of his ancestry; Wuthering Heights refers to Heathcliff as a "gipsy brat" and never addresses who his parents might be. Even Emma plays into this - until the very end, Harriet Smith's origins are totally unknown and that mysteriousness is a key aspect of her subplot.
Mr. Knightly would likely have handled the matter by gathering some local toughs, going to the encampment, and telling them to move on: without a police force, this kind of thing would have been a more personal matter. Romani groups were dependent on the tolerance of landowners, which they attempted to keep by not staying too long in one location (which unfortunately also made them easier targets for accusations).