Noun phrases can be said to have generic reference if they refer to the class of objects represented by the head noun as a whole rather that one or a subset of members. Thus in:

The tiger is a fearsome beast
My tiger is a fearsome beast

The first example is generic in that it refers to the whole class of tigers whereas the second is non-generic in that it refers to one specific tiger.

Genericness interacts with the notion of specificness. Specific NPs are not generic and generic noun phrases are by definition non-specific (although non-specific noun phrases are not necessarily generic). Some grammarians consider genericness to be a special case of non-specificity.

Since genericness refers to a class of species as a whole, the distinction between definite/indefinite and singular/plural becomes less important. Generic noun phrases involving count nouns can be observed with the definite, indefinite and zero article:

The tiger makes a good pet
A tiger makes a good pet
Tigers make a good pet

But we should note that in non subject positions, only the definite article tends to retain its genericness:

I am studying the tiger
I am studying a tiger
I am studying tigers

However, a good deal of genericness depends on the interpretation of the sentence and in particular the semantics of the predicate. Example 3 above could be generic and refer to the class of tigers and equally the first example could refer to a specific tiger that had been mentioned before.

Non-count nouns can be generic only when determined by the zero article, but the noun retains its genericness in all positions:

     Velvet is a good material to work with
     I like to use velvet.