The lineage of the Elven dialects:

                                               Early Elvish
                                    |                             |
                                 Quenya                        Avarish
                     |              |___________________
              Quenya of Aman        |                  |
                     |             Sindarin            |
                  Telerin                            Nandorin

Usually the term 'Elvish' is in reference to Quenya, since it is the most modern and most derived from of the Elven dialects. If you look at the tree of the different 'races' of Elves, it looks (not surprisingly) very similar to the this tree. The most common language spoken in Middle-Earth was Sindarin, as it was the native language of the Grey-Elves of Beleriand. The king of these Elves (Thingol) even went so far as to forbid the use of Quenya, or High-Elven, in his realm. Quenya was the language of the Elves that remained in Valinor after the Oath of Fëanor.

If you're wondering why Avarish is all alone over there, it's because the Avari were the Elves that never went to Valinor, and thus were separated from all the other races of Elves. Once again, a parallel to the race tree.

After about the Third Age, most Elven dialects evolved from Sindarin.