Oh boy I can answer this from an American perspective! I am currently about to graduate with a dual bachelors in History, and one in Anthropology with an Archaeology track!

So Ill start off by just giving the basic 'duty' of each of the three fields you asked.

History: An historian is someone who reads RECORDED history (through primary/secondary sources) and analyze what they've read from any perspective they choose to do so (How such and such fits into the wider literature). For example, Conrad Totman was a very famous Japanese historian from Yale who often wrote environmental historians, e.g., that the terrain of Japan forced the Japanese to employ certain practices (Japan's terrain is extremely rough and mountainous outside the Kanto plain). You can learn that there are also environmentalist anthropologists and archaeologists as well (I am somewhat of one myself).

Anthropologists: An anthropologist is someone who, simply put, studies humans ('anthro' is Greek for human). This is generally (in America, at least) broken down into four major fields: Linguistics, Biological (primatology), Archaeological, and Cultural Anth. Anthropology itself is generally holistic. The big difference between anthropologists and historians is that anthropologists are the ones who study pre-history. This is why anthropology is particularly dominant in the Americas, most of the North American native tribes did not establish a WRITTEN language until the arrival of Europeans. You cannot conduct proper historical research without written documents to draw on. And so this is where archaeology comes into play. That being said, this is mostly relevant to archaeology. Cultural anthropologists study human cultures through field work, as do linguistic anthropologists. Biological anthropologists generally study humans or other primates to help establish the links between us and our ancestors.

Archaeologists: Archaeologists study human history through material remains. A lot of what can be established about a culture without any written (or with one!) language can be established through the collection of material goods left behind. This isn't limited to things such as pots, or spears either; many archaeologists draw research upon human waste left behind.

So whats the final relationship(s) between them all? Well, there are tons. For an example, archaeologists dig up the remains, and if it relates to a period of time where there is written history available, then the data collected will be mostly dealt with by historians. If it comes from pre-history, then the data will be used anthropologically. Aside from digging, many archaeologists also do lab work in which they make their own assumptions about data. This is why in Europe archaeology is usually a track listed under a History degree, rather than anthropology. So yes, they do work together quite often, though not always. The big distinction is whether a topic is dealing with pre-history, or history. Keep in mind also that every human culture has had a pre-history; the Greeks did not always have a written language, they were just the first in Europe to develop one.

Edit: Here are some intro level books I was assigned to read by professors throughout college, if you're interested enough.

Archaeological Theory: An Introduction by Matthew Johnson

Social and Cultural Anthropology: An Introduction by John Monaghan and Peter Just

History: A Very Short Introduction by John H. Arnold