Historically, cow refers to a female, and steer or bull refers to a male. The plurals of these are cows, steers and bulls. The 1896 edition of Webster's Collegiate Dictionary (found on Google books) defines cow as:
- The mature female of bovine animals.
- The female of certain large mammals, as whales, seals, etc.
If you want to refer to more than one of this kind of animal, and don't want to specify the gender, you call them cattle. Cattle is often treated as an uncountable noun.1 To specify three of them, you would say three head of cattle.
There is historically not a singular, non-gender-specific word for one head of cattle. Your father and grandfather used cattle as a singular to fill this gap. Other people are now using cow for this, and this usage is common enough to have made it to the dictionaries. I don't know whether it's common enough to be considered correct among farmers, however, or whether it's just us ignorant city-folk who use it.
1 Update: Looking at Google Ngrams and books, I was surprised to find two cattle used instead of two head of cattle relatively often, although two head of cattle is the more common term.