It depends on where in the world you're looking, and to a lesser extent how exactly you define cheese - there are hard cheeses, soft cheeses, cheeses which be eaten on the same day you make them, and cheeses which take months or years to mature.

In my own area of research, Irish people made and ate a lot of cheese - they possibly ate a lot more cheese than meat (they certainly ate a lot more dairy as a whole, and there's a metaphorical phrase translatable as white meat to refer to dairy products in general). Old Irish law texts are fairly explicit about cheese and what kinds could be served to whom. There were seven clear types: fáiscre grotha (probably "pressed curd"), tanach (which was at least solid, and maybe quite hard), maethal (a soft cheese), grús (maybe a hard cheese as well), táth (probably soft cheese again), millsén (very soft, maybe even nearly liquid) and mulchán (a hard cheese, of which some later accounts claim you need a hatchet to cut it).

There was also something which might or might not fall into the category of soft cheese, a "thick milk", of which the 12th century Vision of Mac Conglinne (a poem about food; a satire on the raptures of religious poetry) describes the varieties: "of very thick milk, of milk not too thick, of milk of long thickness, of milk of medium thickness, of yellow bubbling milk, the swallowing of which needs chewing". This continued up to the 17th century, and Lucas (see Sources, below) quotes a chap called Moryson, in the early 17th century, saying, "They feede most on Whitemeates, and esteeeme for a great daintie sower curds, vulgarly called by them Bonaclabbe". Bonaclabbe is a crude Anglicisation of bainne clabair, again, "thick milk".

There was much less regard for cheese in England, where it was regarded, more or less, as what you eat when you've completely run out of other foods. Surrounding nations have always regarded the English as devoted carnivores, though - see the French slang term rosbif, used right up into the 19th century, and even beyond (it has even had a recent renaissance in anti-English graffitti in France, although that comes under the 20-year rule). Oddly, though, in the recent lacuna in cheese production in Ireland, cheese was imported from the UK (see this answer of mine from a few weeks ago for more details). There are also theories as to why cheese was not well regarded in England, to do with the replacement of farming populations, and the loss of cheesemaking skills - good cheesemaking is an intensely local process, because of the local airborne bacteria, and the same methods that make good cheese in one place will not necessarily do so elsewhere. But that then gets into the debate of the invasion-and-replacement theory versus gradual cultural change theory with the arrival of the Anglo-Saxons, the Vikings, and the Normans, and I'm not able to speak to that in any detail.

Nevertheless, Andrew Boorde, writing in 1542, says "Cheese is made of mylke; yet there is iiij sortes of chese, which is to say grene chese, soft chese, harde chese, and spermyse. Grene chese is not called grene by the reason of colour, but for the newnes of it, for the whey is not half pressed out of it; and in operacyon it is colde and moyste. Softe chese, not to new nor to olde, is best, for in operacyon it is bote and moyste. Harde chese is bote and dry, and euyll to dygest. Spermyse is a chese the which is made with curdes and with the iuce of herbes : to tell the nature of it, I can not, consyderynge that euery mylke-wyfe maye put many iuces of herbes of sondry operacyon & vertue, one not agreynge with another. But and yf they dyd knowe what they dyd gomble togyther with-out trowe compoundynge, and I knowynge the herbes, then I coulde tell the operacyon of spermyse chese. Yet besyde these iiij natures of chese, there is a chese called a rewene' chese, the whiche, yf it be well orderyd, doth passe all other cheses, none excesse taken." So there's plenty of variety of cheese in England, too, and lots of local variation.

Generally speaking, though, cheese has been a good strategy to use up excess milk - and at some times of year, particularly just after calves have been weaned in mid-to-late summer, there is a lot of excess milk from cows. Similarly, a goat in high summer can produce up to four times what she will in leaner seasons, so there's milk to use up there too. And while the return, weight for weight, of cheese from milk seems low (generally about 1kg of cheese for each 8-10l of milk), the leftover whey has many uses, and a kilo of cheese in February or so is worth a lot more than 10l of excess milk in July. And obviously, the cheese will keep, while the milk won't.

In terms of ancient history, I can tell you that Pliny the Elder mentions cheese in his Historia Naturalis, and it probably came to Rome from Asia Minor, where it was made by nomadic peoples. Also, Homer mentions cheese made from the milk of sheep and goats, possibly similar to modern feta, and Aristotle mentions cheese made from the milk of horses and donkeys.

TL;DR: Cheese was not rare, people ate lots of it.

Sources:

Cathal Cowan & Regina Sexton, Ireland’s Traditional Foods

A. T. Lucas, 'Irish Food Before The Potato', Gwerin vol. III No. 2 (1960) 8-43

Robert Smith, The Great Cheeses of Britain and Ireland

Peter Brears, Cooking & Dining in Medieval England

Andrew Boorde, The fyrst boke of the introduction of knowledge made by Andrew Borde, of physycke doctor. A compendyous regyment; or, A dyetary of helth made in Mountpyllier