You will hear this expression mostly in connection with food or beverages. It refers to a food/beverage that most people likely will not appreciate the very much the first time they taste it but over time and with increased exposure one might – or might not – acquire a taste for it.
The reason why one would need to acquire a taste for something generally is because it is unfamiliar or strong in one way or another, it could be taste, odor or even texture.
For example: I never acquired the taste for coffee. To this day I can’t drink it unless I put so much sugar and milk in it that it becomes unrecognizable – but then it isn’t coffee. Examples for acquired taste are many: asparagus come to mind, alcoholic beverages, capers, caviar, olives, things like that.
When I think of odor I have a few French and German cheeses in mind that taste very well – if you can get yourself to put them in your mouth, that is, because of their strong “stinky-feet” odor. Durian, the fruit that has an odor somewhere between “gym-socks” and rotten onions is another example.
Texture: oysters. I know people who flat-out refuse to even try oysters because they are slimy. Passion fruit might be another, it smells so delicious and then you open it and see but a few seeds in some gooey greenish liquid.
So next time an American friend or co-worker asks you to try something with the qualifier that it is a bit of an acquired taste you know what you are in for.