Archaeology is the evidence of history. Of course we know that history exists, but archaeological evidence is what gives that history its shape and color. 

It is of most importance for learning about prehistoric societies, when there are no written records for historians to study, making up over 99% of total human history, from the Paleolithic until the advent of literacy in any given society.

For example, we may have written or other documentation which suggests that a certain people group once inhabited a land, and we may know (or think we know) how they must have lived based on similar groups or other some kind of external evidence. Until we see the artifacts of that culture, however, we really cannot be certain of anything. 

Artifacts reveal much more than the existence of a people group. The tools they used can tell us how they built things; the foundations of their homes (whatever they looked like) can tell us if they lived in large or small family units. The weapons they used can reveal what they hunted for (both human and animal), and the cooking utensils they used can tell us what they ate and how they cooked it. The possible list of archaeological finds is virtually limitless, including everything from clothing, jewelry and other possessions to art and other cultural expressions. Religious artifacts, whether in the form of art or of actual artifacts, can also reveal their religious beliefs and traditions.

While these are all interesting things to know, they are also valuable in connecting the dots of history. How and where people groups migrated can often be determined by archaeological finds; why a culture died out is also knowable, at least in part, by their artifacts. Even disease patterns, such as plagues, can be traced through archaeological finds, as well as animal migrations.

Every time archaeologists make new discoveries, as evidenced in some of the answers above, new revelations about history are made. Some of these finds answer questions; others pose new questions or cause us to rethink old theories. In any case, archaeology is the evidence or proof of history, especially before written records were kept, and therefore the two are irrevocably linked. 

Archaeology allows us to discover, uncover and recover objects and materials which often support or possibly disprove historical records or accepted beliefs attached to previous eras and events. Archaeologists can attach some kind of chronological order to events. In prehistoric archaeology, it is crucial that there are items which can allow us to trace the past as these  archaeological finds become the evidence to support otherwise verbal accounts of history. The cultural value of archaeology is also incalculable and archaeologists work with anthropologists in unearthing not only human remains, buildings, tools and other objects made by man, the main function of the archaeologist, but also the environment, climate and culture surrounding the archaeologists' find which is the main field of the anthropologist.

In South Africa, the discovery of remains, affectionately called "Mrs Ples," in 1947 changed the way modern man perceives the past. Paleontologists, who study past species as opposed to man-made objects such as archaeologists study, discovered the skull, throwing light on the theory that mankind originated in Africa. Despite the small stature and small brain of Mrs Ples, controversy still surrounds the debate as to whether Mrs Ples is male or female, there is no doubt that "she" walked upright, changing the perception of time and man's evolution.

It was archaeologists who were digging at Sterkfontein caves in South Africa, looking for evidence of man's presence there through their tools, bones left over from meals, remains of the pots they cooked in, etc that ultimately led to the discovery. Although there is still discussion on the first location of human beings, homo sapiens, themselves, no one disputes the valuable contribution to the understanding of the past that archaeology, anthropology and paleontology bring to our understanding.  

Seeing is believing.

Archaeology deepens humanity's understanding of past historical events by providing rich details about how people lived, their daily routines, culture, and society, or even help differentiate between fact and fiction.  For example, many scholars and historians believed Troy to be a mythological city in Homer's Iliad--until Frank Calvert and Heinrich Schliemann's excavation of Troy.  The discovery of the ruins by these archaeologists added serious perspective to what historians already knew about the Aegean Sea, trade, and warfare during the Bronze Age. 

Calvert and Schliemann's discoveries concerning Troy provided the hard evidence to fill in those missing blanks in the historical record.  By delivering much needed proof, answers, or evidence, archaeological discoveries continue to challenge the way humanity engages with history.

Anthropology and archaeology both are the venues of history. By studying ancient peoples and their artifacts and structures, man learns much of what has entered into his own humanity. Certainly, much has been learned from the discoveries of the products and the remains of man from previous ages. Just recently, reports in London that skeletal remains of Black Death victims having been discovered in an ancient burial site has shed new light on the accepted idea that rats spread the plagues of the 14th and 15th centuries. The site which was unearthed in the excavations being made for the new Crossrail in England's capital city revealed the skeletons of the victims of these pandemics. For,now scientists have determined that rats were not the cause of the spread of the disease; instead, studies of the DNA reveal that coughs and sneezes were the cause of the diseases, rather than the fleas on the rats as previously believed.

Many archaeological finds have been the source of beauty, as well. The decorative artifacts of the Ming Dynasty, the Greeks, and the Romans have been inspirational to those whose civilizations have followed. Certainly, the Neoclassical era and the Renaissance owe much of their art to the discoveries of ancient efforts. Anyone who has visited the Roman Forum or Pompey and walked around the edifices and artifacts that stand as witnesses to a time before Christ cannot but be moved by a beauty and movement that yet communicates with man. Indeed, who has stood before her in the Louvre and not been inspired by Winged Victory that once lay buried under the sea?  

Archaeology and history are completely interlinked.  Especially when studying ancient cultures and civilizations, most of what is learned comes from archaeological expeditions that succeed in uncovering artifacts and human remains from those earlier periods.  By finding and studying -- including the use of carbon dating and other methods to determine the precise age of the items discovered during archaeological digs -- buildings, cisterns, roads, tools, weapons, and other items buried or lost for hundreds or thousands of years, a more complete and accurate picture of the people who lived in a given location can be drawn.

Archaeology is not only vitally important to the study of ancient history.  It is used today to gather information and evidence for use in criminal trials of people suspected of carrying out heinous acts, especially former dictators whose efforts at hiding evidence of their crimes -- particularly, the mass murder of political opponents -- are increasingly defeated through modern imaging technologies used by archaeologists and others to locate such hidden sites.  Following the end of the long civil wars in countries like El Salvador and Guatemala, human rights organizations used archaeologists to dig up mass graves for the purpose of gathering evidence against the former dictators responsible for the deaths.  Skulls of adults and children with bullet holes and evidence of bound wrists provide conclusive proof that atrocities occurred, which are important not just for future trials, but for the accurate recording of history.

Archaeology is similarly used today in Israel and the Palestinian territories by both the Israelis and the Palestinians, as each seeks to gather physical evidence that their ancestors were there first and, consequently, the rightful owners of the land.  

The most important intersection of archaeology and history, however, lies in what archaeology has told us about civilizations around the world, including the Incas, the Mayas, the Athenians and Spartans and Romans and Jews, and many others.  By uncovering the remains of ancient temples and homes, and by studying the earthen jars and cookware of those ancient civilizations, historians are able to present far more accurate pictures of what the world looked like before most recorded history was written.

Archaeology can play a very important role in the study of history.  It is, of course, more important for the study of prehistoric times, but it can be important even when we study more recent events.  Since you have tagged this with “archaeology and Jamestown,” I will refer to Jamestown in my answer.

Jamestown was founded in 1607.  This was, of course, well within historical times.  Therefore, we have some historical records about what happened in the early years of that settlement.  However, those records are fragmentary at best.  In addition, even the best of records would not tell us everything about life in a settlement such as Jamestown.

Archaeological finds tell us, for example, about the daily lives of people in Jamestown.  These sorts of things would not have been of much interest to the people then.  They would not have kept records about things like that.  But archaeological evidence tells us, for example, that there was a fair amount of economic activity going on in Jamestown.  It also tells us that there were probably friendly interactions with Indians early on.  These are things that cannot be gleaned from the records from 400 years ago.

Because records from some historical times are not complete, and because even complete records do not capture the mundane details of daily life, archaeology has an important part in the study of history.