You use fewer with countable nouns such as students and sort of.

You use less with uncountable nouns such as water.

few, fewer, fewest, little, less, least are the degree of comparison.

So the answers of your questions:

The correct one is:

1. There are fewer students in class today. (Students = Countable Noun)

2. There is less water in the lake today. (Water = Uncountable Noun)

"fewer" = not as many, "less" = not as much

Fewer coins, less money.
Fewer minutes, less time.

It's not something you need to worry about in daily speech. People often use "less" instead of "fewer".

I remember in the past, Tesco (a supermarket) had to change a sign that said "10 items or less" because of some linguists.

Academically speaking, I don't know the correct gramatical reasoning, but I can try to explain it in basic terms. 

Fewer is a comparison word, and so is less. However, fewer is used when the object being compared or analyzed is plural, and less is used when the object is singular. Students are plural, so fewer makes more sense. However, water is one entity, so less is used in that case.