Initially the equites were simply those property classes which could afford provision of a horse, either publicly-provided or privately-owned. This included both patrician centuries and wealthy plebeian centuries. By the middle Republic this had been totally superseded, and by the late Republic it was ancient history. There's some debate over what exactly constituted the equites at the end of the Republic--there was a late 2nd Century plebiscitum reddendorum equorum, which obviously has to do with the supply of public horses, but its precise meaning is unclear. In any case, during the late Republic the equites were sort of a semi-official sub-order of the plebs. Very little distinguished them from the senatorial class. Like (the vast majority of) senators they were plebeian and they held pretty much the same amount of wealth--under Augustus the equites were those valued at 400,000 sesterces. What distinguishes the equites is that they are not politically active, at least not in senatorial magistracy and other elected office. It's more complicated than that, but basically the equites were those comparable to the senatorial class in wealth (and morals, as they had to be confirmed by census) but who had no aspirations to senatorial involvement. Under Augustus and in the Principate this non-senatorial identity becomes pretty defining, as equestrian magistrates developed under the emperors, positions totally independent of election and the senate--these sorts of magistracies reserved for those of non-senatorial rank were totally unknown in the Republic