the War of the Roses was between two Branches of the Royal English house (York and Lancaster), so it would definitely not be seen as a war between either peoples or countries. A better example would be the hundred years war, which began as a war between Edward III of England, and Philip VI of France, over who had the better claim to the french throne. The war is (and was) viewed as a war between the two royal persons, and not between England and France per se. Edward actually viewed himself as being French, and the Great war between England and France was a concept that arose several years after the death of Edward. for more information on this, i recommend "The Wars of Edward III" by Cliff Rogers. The book is a collection of sources form the beginning of the hundred years war, and here, the War is justified by Edwards personal claim on the throne of France, and thus, can best be described as a war between two men. Like Pugnacity points out, the idea of a war between peoples didn't really exist before the rise of nationalism during the 19th century, and the idea of a war between countries is not much older, since most countries were viewed as the personal fiefs of a given monarch.

As is often the case when discussing the Middle Ages the answer is really: it depends. Somebody has already pointed out that the War of the Roses is a weird case so I'll leave that example aside and talk about two well known medieval wars: the wars of Scottish Independence and the Hundred Years War.

The Scottish War of Independence (1296-1328, although depending on how you define the conflict you could expand those dates some) were fought between Edward I and various nobles of Scotland most famously William Wallace and Robert Bruce. I'll spare you the full lecture on how the war came about and skip right to the relevant bit. The Declaration of Arbroath was an open letter written to Pope John XXII by members of the Scottish nobility. It includes a rather famous passage that goes like this (translated from Latin of course): "...for, as long as but a hundred of us remain alive, never will we on any conditions be brought under English rule. It is in truth not for glory, nor riches, nor honours that we are fighting, but for freedom – for that alone, which no honest man gives up but with life itself." That's some pretty nationalistic stuff there. This declaration probably doesn't reflect the opinion of all Scots but it certainly was an opinion held by some that suggests that a certain part of the Scottish nobility viewed their enemy as the English and not just Edward II (Edward I died in 1307 leaving his son to try and suppress the Scots...he failed). Geoffrey Barrow has a very readable biography of Robert Bruce but be warned he was a card carrying Scottish Nationalist so he's inclined to overemphasize those elements. Michael Prestwich's book The Three Edwards includes a succinct but less biased (or biased from a different perspective depending on your view) account of the Scottish wars.

With the Hundred Years War, though, we have...well a staggering complicated conflict. Let's just stick with the first part fought primarily by Edward III and his son The Black Prince. This conflict was primarily portrayed as a war between Edward I and at first Philip VI (until 1350) and later John II (until 1364). This wasn't really a conflict between two kingdoms because the main point of contention was that Edward III thought that he should be the king of France and not Philip.

Whether Edward III had the correct claim or not..is messy. According to Salic Law he certainly did not but he could argue that via Roman Law he did since he was closer in relation to the deceased previous king (Edward III's mother was that king's sister). What law applied in this situation is...often unclear. Interestingly a very similar debate played a large part in kicking off the crisis that led to the Scottish Wars of Independence. The French went with the Salic interpretation and Edward III at the time was in no place to dispute them. Edward II had only been deposed the previous year and Edward III was stuck as a bit of a puppet king to his mother and things weren't looking so great for the power of the English king at that moment (deposing monarchs tends to cause problems).

Anyway, that's all a very long way of saying that Edward III was fighting for the right to be king of France so it would be strange to view his war as between the French and English kingdoms, according to Edward he was in charge of both. That said, it could be argued that Edward didn't really believe he'd be the King of France and largely used that as an excuse to wage war. It's also unclear how much the common soldier would care about things like what the war was about. Being a soldier on a successful military campaign could be hugely profitable and it's possible what they cared about was money not some vague concept like 'nation.'

Michael Prestwich's The Three Edwards applies here as well as in my previous example. M.H. Keen's England in the Later Middle Ages also covers these events pretty well. For a primary source Froissart is probably the best account of the first period of the Hundred Years War.

If you were to look at different wars in different regions across Europe you would find plenty of different examples. Italy featured extensive warfare between city-states using mercenary armies, for example.