Hello there,
Very basically transistors are amplifying devices. They take a small signal of some type and amplify it to a larger amplitude version.
Because of their ability to amplify, they also function as switches, being able to switch a larger current with a much smaller current or with a small voltage.
Also because of their ability to amplify they are used in other applications too, but amplification and switching are the most common.
There are other applications that take advantage of the transistors more esoteric characteristics, but these are more rare.
The main specifications are the current through the output circuit and the voltage across the output leads. That limits the usefulness in a given application.
There are other spec's too that come into play such as switching speed, and others that may be important to the application.
Whenever i see someone who wants to understand transistors for the first time i always recommend that they start with the bipolar transistor used as a switching transistor. That allows the person to learn how the basic control of the transistor works.