Hmm, ok, I'm guessing that this question is emerging from the Latin words used for each of these groups, which have all been translated as 'knight' in English. It can get quite confusing.

Members of the Roman equestrian order were equites. Literally, the word means horseman, and it comes from the role that members of this class were expected to fulfill in the early Roman citizen army. Eventually, it became a social rank (the second highest in the Roman world, after the senatores), separate from any connotations of military training, expertise or service. The word was translated as 'knight' by many early-twentieth-century writers, although it would not be nowadays, as it gives completely misleading connotations. I'm not a Romanist, however, so you'll have to wait for one of our resident ancient historians to come along if you want a more detailed explanation of how they fitted into Roman society at various points in time.

Medieval knights were called milites. Miles was the Classical Latin word for soldier, and the word retained this meaning broadly throughout the early middle ages, being used primarily by authors in a military context, although it gradually began to pick up more positive connotations too (the theology of a miles Christi, a soldier of Christ, i.e. a monk, was quite popular in the 9th century, for example).

This changed in the middle of the tenth century however, when people began to use miles as a title or a personal description. This seems to be related to wider changes in society, in which military activity was seen to be an exclusively aristocratic phenomenon, so by adopting being a miles as part of their identity, these men were asserting that they were part of the nobility, rather than simply free men. Further supporting this idea is that fact that the title first appears to have been used only by the lower aristocracy; counts, viscounts, dukes etc. (and their families) don't start to call themselves milites until much later, usually around the end of the eleventh century. It is miles used in this context which historians usually feel comfortable translating as 'knight'.

From the eleventh century onward, 'knights' gained more and more connotations which remain with the word today, including honourable behaviour, obligations to women, the church and the poor and other values we call 'chivalry', which gradually formed into a sort of recognised code of behaviour which drew on older expectations of the correct behaviour of an aristocrat. The institution also became more formalised; we see examples of young men being 'made' into knights in a formal way by the beginning of the twelfth century, usually by being presented with a sword and belt - the first text for a church-based dubbing liturgy dates from the early thirteenth century. By the mid-thirteenth century, it's very easy to see knighthood as a real institution, with big celebrations being thrown for the formal knighting of a young noble. In the thirteenth century, orders of knighthood began to be established, most famously the Order of the Garter, which still exists today. Towards the end of the middle ages, knighthood began to lose some of its more militaristic connotations and became the expected status of any non-cleric male noble, although this was a slow process, and wasn't complete until well beyond the end of the medieval period.

Could explain a bit about what led you to ask this question?