Avocados do not ripen on the tree, that is, they do not get soft while on the tree. Once you pick an avocado, it takes about 7 to 10 days for it to soften when left at room temperature. You can speed the process up slightly by placing the avocado in a bag or slow the process down by keeping the fruit in the refrigerator.
It is hard to tell from the outside when an avocado is mature. Commercial growers conduct a "dry weight" test which gives an indirect measure of the oil content of the fruit. If the oil content is too low, the fruit is not ripe yet and will shrivel or stay rubbery instead of getting soft.
The Hass avocado typically ripens in February and is good through June or July. These dates depend a lot on where you live and climate conditions. Some years you can pick larger fruit as early as December and they will ripen up. In Ventura County, fruit can remain on the tree and still be good into August and September.
Varieties that ripen other times of the year (dates based on the experimental plot in Irvine, CA):