Speaking for linguistics; The Battle of Hastings is the sole event that made the English language what it is today. The successful invasion of the Normans (French speaking peoples) caused their language and culture to collide with that of the Old English (Germanic) speaking Saxons.

In the long term this would result in the horribly confusing mix-mesh of Romantic and Germanic languages we call English. Had King William not been successful, words and culture of England and all the lands she would go on dominate would be very different. It's a change so drastic that the impact of it not happening is almost impossible to visualize.

This is one of the main reasons 1066 is talked of so much. Saying Britain was forged on that battlefield is not an overstatement. And after the battle was over William built a tower in a place that would become the capital for a quarter of the planet.

It moved Englands focus from Scandinavia to the continent. When the normans took over the Scandinavian kings either backed down or were dead. Because English kings became also rulers to big principalitys in France the French king could take these without upsetting his other vassals. This was a boost to centralisation and basically created the ground where Francis, Richelieu and the Sun king build their house on. If the normans had failed the Danish may have tried and succeeded. Before Hastings there was a North sea Axis between England and Norway/Denmark. After Hastings there was a North South Axis between England and France

Well, for one, it changed England's language from German-based Anglo-Saxon to a fifty-fifty mixture of it with Latin-based French - the French have a lot of mythology about their "Frankish" kings, but it was just a top layer that didn't even penetrate past the northeastern corner; the bulk of the population remained Latin-speaking, only a few words like "bleu" instead of azul and "Francais - Frankish" most prominently.

This caused a lot of cultural changes, especially a more feudal system a much more sharply divided hierarchy than before. Land ownership changed a HUGE amount, even now, the switch from a lot of people owning small amounts of land to a few people owning huge land grants is still true. It embroiled the English people in a ton of wars with France - the Hundred Year war, the marriage of the Duchess of Aquitaine.