In the Roman Middle Republic equestrians were a social class consisting of men wealthy enough to serve as mounted warriors. They served to the point of being nearly wiped out in the Second Punic War, a war during which they switched to heavier gear (though certainly not as heavy as later knights).
The equestrians were a citizen cavalry, not as highly armoured and well-trained as knights, but we're talking 1000 to 1500 years before the Medieval knights. Professional soldiers did not exist at that time! Now I* wouldn't call those equestrians 'knights' in the Medieval sense, but they were a social class involved in mounted warfare well before the stereotypical knights ever existed.
Of course later on in the Roman empire the equestrians lost much of their military function, the cavalry being supplied by allies or (even later) hired tribes. So there is not a direct line from the early Roman equestrians to the Medieval knights.
*They often are called 'knight', as people apparently consider 'equestrian' not an English word or something.
(This answer has been edited to remove references to deleted comments)
A caveat before we begin. This is one of the most argued about, and least clear, issues in medieval history. It is an issue precisely because the early years of knighthood occurred during a period in which very little writing was being done. Much of what we know of it comes through the lenses of clergymen. Thus, bear in mind that this, while based on heavy personal research, can hardly be called definitive.
The knight as a distinct class or profession really cannot be attested to prior to the 10th century and the Carolingian breakup. Documents from this period refer to them variously as miles or eques, both obviously Latin terms, the former denoting a soldier, the latter a cavalryman. At this point, knights were in all likelihood not particularly prestigious. Valuable and relatively high status soldiers, yes; but not a title that was synonymous with lordship and aristocracy. At this point, a proper European nobility does not exist; so knighthood can hardly have been co-opted by them. Best guess as to equipment and fighting style at this time is an armored, but not fully armored man (a chain shirt, coif, and helmet), who can fight as a highly mobile skirmisher (the lance had probably not reached full development, and was a dual-purpose throwing/stabbing spear), a shock cavalryman who closes with the sword, and an elite heavy infantryman who can stiffen a line or shield wall.
The classical conception of knighthood begins to take form in the 11th century, especially in the last half. It is during this century that knighthood becomes popular with the aristocracy of Europe. Where before this time knights and lords were distinct from each other, the lines begin to blur as lords have themselves knighted and embrace a sort of double status of aristocrat and knight. William the Bastard, while still teenage duke of Normandy, has himself knighted in the 1040s. At the same time, this seems to have increased the status of simple knights, and they became permanently a part of the aristocracy proper. Equipment and fighting techniques are not awfully different from the 10th century, though more emphasis on the lance as a shock weapon may have developed.
The 12th century is really the age of knighthood in many respects. It is during this century that a concept of chivalry develops, especially in southern France, emphasizing valor, loyalty, humility, piety, etc. This is heavily bound up with the rise of the troubadours and a general rage for poetry among the aristocracy. Chivalric romances serve not only as a popular literature among the knightly classes, but a way of binding aristocrats together with a common ethos that separates them from other classes and emphasizes their own special and unique nature. It is important to note that chivalry was not a fixed ideology, and nor was it adhered to at all times, its various tenets being violated when necessary or expedient. At least one historian, David Crouch, has argued that this class ideology of chivalry explains the distaste for non-knightly mercenaries that colors the chronicles of the late 12th and early 13th centuries (which were themselves written by men from the aristocracy; many of them were clerics, but they had been exposed to knightly concepts in childhood).
At the same time, the 12th century is really the beginning of the tournament as we recognize them. These had been occurring in some form since at least the 11th, and may have been the descendant of Roman cavalry games, but in the 12th century they exploded. Powerful men throughout Europe, but especially in modern northern France and western Germany, hosted these military games. Lords and their household knights, groups of landless young knights, and individual men ranged around Europe, moving from one tournament to another, hoping to acquire wealth and status and, in the case of the poorer sort, land. At this point, the tournament was not the highly stylized joust. It was a mock battle, fought over a fairly vast area (it often included several villages and the surrounding fields and woods), in which groups of knights, often from the same region, competed to disable or capture other knights in order to take their horses, armor, and weapons and hold them for ransom. They were often hideously dangerous, scarcely less so than war.