You ask a broad question that might be interpreted in a number of ways – most obviously, perhaps, "What happened to the Saxon survivors who participated in the Battle of Hastings?" and "What happened to the Saxon people of England after the Norman Conquest?" I'll try to touch on both these problems in this answer.

Scholars have invested considerable energy in attempting to determine how large the armies that met at Hastings were, and while the solution to this problem is ultimately unknowable, estimates of Saxon forces cluster at around 8,000. The battle was unusually fiercely fought and unusually protracted by contemporary standards – the action began early and lasted for most of the day – and while estimates of casualty rates are even harder to estimate with any accuracy, it has been suggested (based on extrapolations of deaths among known, hence noble, participants) that anything up to half of the Saxon army may have been killed or died of wounds. Certainly this number included not only the Saxon king, Harold, but also two of his brothers – Gyrth and Leofwine.

Hastings, thus, substantially thinned the ranks of prominent Saxons who might realistically have posed a threat to the Norman victors, but while popular perception of the Conquest is that it was achieved as a result of a single, cataclysmic battle, this is a considerable over-simplification that scholars of the period have devoted a good deal of time and trouble to nuancing. To begin with, the Saxon royal council – the Witangemot – preferred to elect a new king, Edgar Atheling, to capitulation to William, and he was proclaimed as Harold's successor in London. The capital itself refused at first to capitulate to the advancing Normans, and did so only after more than a month of further skirmishing and a concerted Norman push to threaten and encircle it. Edgar – still a boy of 14 or 15, too young to lead an army in the field – surrendered and went into genteel imprisonment in Normandy for several years, joining another of Harold's brothers, Wulfstan, there.

Victory at Hastings, moreover, left William and his own surviving men with an entire country, and a population estimated at roughly 1.5m, to subdue. There is no doubt that the Conqueror received substantial reinforcements from his duchy in the aftermath of Hastings, and – once king – he also had the vast advantage of substantial parcels of former Saxon royal and noble lands to hand out to his followers. Nonetheless the entirety of the Norman forces (perhaps 20,000 men at most) were insufficient to do more than hold parts of the country down at first. Motte-and-bailey castles – quickly constructed and easily defended against guerrilla attacks – were key to this phase of the conquest, but Normans venturing beyond their walls had to travel in large groups for some time, and solitary incomers risked ambush by Saxons for much longer than that. While the progress of conquest and pacification is not easy to follow in fine detail, we need to remember that there were serious Welsh raids, invasion from Scotland, pinprick seaborne attacks from Harold's sons (who had fled to Ireland), and a major Saxon rebellion (led by surviving Saxon nobles such as Edwin and Morcar, the rulers of Mercia and Northumbria) in 1069-70, which was only put down with difficulty, and resulted in William ordering the infamous scorched-earth tactics known as the Harrying of the North in retaliation. A second significant uprising, in Northumbria in 1075, was combined with more persistent, lower-level resistance in inaccessible parts of the kingdom. The Saxon leader known as Hereward the Wake, who had his base in the marshy fenlands of East Anglia, is the best-known of the latter, though his legend was considerably embellished during the Victorian era.

All in all, it is probably safe to say that it took the Normans a full decade to obtain a firm grip on the Saxon kingdom, but it is equally true to note that William felt secure enough to spend long periods in Normandy from 1072, and that by the time of the king's death in 1087, England was secure and Norman rule was broadly accepted by the conquered Saxons – who had by this time been almost entirely deprived of a significant financial or territorial base from which to mount further resistance. Domesday Book, compiled in 1086, lists only four major landholders with Saxon names, and, even after granting large swathes of land to his followers, William still held more than a third of the land in four English shires, more than a fifth in eight more, and at least one-tenth in practically all the others. This meant that royal wealth was drawn from the full breadth of England, and that the monarch had a greater locus of power in every part of the kingdom than his Saxon predecessors had enjoyed. The impact of all this at a more local, and more peasant, level is very hard to assess, but MT Clanchy has noted that acceptance of (and perhaps even admiration of) the Normans is suggested by the widespread popularity of Norman names for Saxon children in the period 1070-1110. According to Clanchy's figures, the proportion of native children given foreign names in Winchester, the old capital of Saxon Wessex, rose from 29% in 1066 to 62% by 1110.

As for what happened to "the rest of the Saxons" – some, certainly, chose not to live under Norman rule. Edgar Atheling, once released, fled to Scotland, picking up a group of Northumbrian rebels and Danish malcontents along the way, from where he kept up active attempts to gain the throne until the middle 1070s; he continued to meddle in Anglo-Norman affairs as late as 1106. Other groups of Saxons also left the country, not least to take service with Byzantium as members of the empire's famed Varangian Guard. The best known, if also the most mysterious, of these exiles were the group supposed (on extremely flimsy evidence) to have sailed from England in 350 ships some time between 1075 and 1091 to establish a colony known as New England in the Crimea, constructing towns there that they gave the names of "London" and "York".