I've answered a couple of questions about choosing the right case to use (you can find them here and here). In each of these posts, I've been talking about the concept of ainesana, but I haven't really devoted time to explaining what they are and how to recognize them. So here goes!
The word ainesana is composed of two words, aine 'material, element' and sana 'word'. Ainesana literally means an "element word" or "material word", and its English counterpart is the uncountable noun. However, unlike in English, in Finnish it's often less useful to think of these words in terms of counting. In my experience, it's often more useful to think about dividing, and some Finnish grammars use the term jaollinen, 'divisible' to mean the same thing - if you can divide it, it's an ainesana. Ainesanas behave a bit differently than other words, especially when it comes to using the partitive case.
The following groups of nouns are ainesanas:
1) food and drink: kahvi 'coffee', juusto 'cheese', mehu 'juice', leipä 'bread'
2) elements and materials: vesi 'water', nahka 'leather', tuli 'fire'
3) abstract concepts and feelings: rakkaus 'love', ystävyys 'friendship', ilo 'joy'
Ainesanas are things that you can stick into a blender and mush up without them losing their essence. A mashed potato is still a potato, but stick your cellphone (not an ainesana) into your vitamix and what you get is a pile of broken plastic, glass and metal. Love remains love even when your heart is broken.
Another way to phrase the same idea is that with ainesanas, even the tiniest bit of the thing is still the thing. If you pour some coffee into a coffee cup, you now have coffee in a cup as well as in a coffee pot. Cut up a cake, kakku, and dish it out and you have lots of plates with cake. If you cut up a t-shirt in a similar way you end up with garbage, not a t-shirt, but the shards are still fabric, kangas, which of course is, again, an ainesana.
If you can divide it without it losing its essence, it's an ainesana. If you can't, it's not.