If we want to get picky about it, the first monarch of Great Britain was Queen Anne, as the kingdom of Great Britain wasn't formed until the 1707 political union between England and Scotland. (Before this time, the crowns of England and Scotland were generally held by the same monarchs, a practice that started with James I/VI after the death of Queen Elizabeth, but the nations were separate).

But what you're probably asking about is the antecedents of the current royal family, and how far back the current monarch of Britain can trace her ancestry. Traditionally, Alfred the Great (849-899) is regarded as the first king of the English and the founder of the English monarchy, although his grandson Æthelstan was the first to rule over land that roughly corresponds to the modern area of England.

Alfred was king of Wessex, one of several "English" kingdoms in the ninth century that warred with one another and suffered under the Norse invasions of England; he was the first monarch to successfully unite parts of the English kingdoms against the Norse and to successfully organize a united defense against Viking raids, through a system of fortified burhs, as well as building a small fleet that could be used in raids and in defense against raids.

By the time of Alfred's death, he had made Wessex into the strongest of the English kingdoms and also extended his rule into western Mercia, including the city of London (though Æthelred, the lord of the Mercians, ruled in London). Alfred's son Edward (the Elder) succeeded his father as king of Wessex, and was able to add to his lands East Anglia and Essex, as well as all of Mercia after the death of his sister Æthelflæd, who had been married to Æthelred and ruled Mercia after his death.

Edward's son Æthelstan is generally regarded as the first king of "England" as a more-or-less united political entity, adding York and Northumbria to his holdings and negotiating peace with both the Scottish kings and the Welsh (although the peace with Wales was longer-lasting; the Scots proved troubling and there was border strife throughout Æthelstan's reign.)

The British monarchy goes back to... 1707. Before that, England(+Wales) and Scotland were two seperate kingdoms, that happened to have the same Monarch (but, besides the monarch, different governments). That of course, only goes back to 1603, when James VI of Scotland inherited the English throne from his childless, sibling-less cousin Elizabeth I of England (James's grandmother, wife of James V of Scotland, had been the sister of Henry VIII of England).

Now, as to the English monarchy, that traces back to William the Conquerer, who, well, conquered England. He was, admittedly, related to the previous King, Edward the Confessor (Edward's mother was William's great-aunt) but that was only a connection to the English House of Wessex through marriage, not blood. (Harold Godwinson, who did become King briefly after William, was also only related by marriage; he was Edward the Confessor's brother-in-law.)

However, while this was a break from the House of Wessex, it was later patched up. The male members of the House of Wessex almost all died off, but the man who would have been King (Edward the Exile) went to Scotland. His son died childless, but his daughter married the King of Scotland. Her daughter married... Henry I, son of William the Conquerer.

Anyway, so the House of Wessex. They were called the House of Wessex because they were, before the unification of England, Kings of the petty kingdom of Wessex (in southern England). Their most famous member is probably Alfred the Great, who defeated the Danes (who had conquered most of the English Kingdoms). Although he didn't formally rule of England after defeating the Danes (he made peace with the Danes, leaving them with some territory, and there were other English kingdoms, like Kent), his saviour of the Christian Anglo-Saxons from the Danish conquerers made him the pre-eminent ruler in England, and he was known as "King of the Anglo-Saxons".

Alfred's son, Edward, extended his de-facto power over all of the English kingdoms, inheriting Mercia from his sister, and conquered some of the Danish lands. All the Danes submitted, in theory, to his authority, but archeological evidence suggests they were fairly independent in practice, just no longer a constant threat.

The first person who can probably really be called King of England is Aethelstan, son of Edward, as he conquered the last Danish kingdom (York), gained the crown of Northumbria, and thus united all England under one King. He was also acknowledged as suzereign by the King of Scotland and the Princes of Welsh (note: this is not the same as saying he was King of Scotland and Wales, he was not).

But you can see, that saying who was the first King of England (unlike the first Queen of Britain, which was Queen Anne) isn't an exact simple answer, because England isn't something that just sprung into being out of nowhere.

Scotland's early history is similarly complicated, but I'll let someone who knows more about Scottish history explain about that.

But, to answer your question directly (with England, rather than Britain, because I think that's really what you're asking); Yes, it was just a family (the House of Wessex) who gained more and more power and land until they ruled all England, about 1100 years ago. They were then replaced in 1066 by the House of Normandy, whose descendants (who are also descendants of the Scottish Kings, slightly more directly actually) still rule today.