BJTs, FETs and MOSFETs are all active semiconductor devices, also known as transistors. BJT is the acronym for Bipolar Junction Transistor, FET stands for Field Effect Transistor and MOSFET is Metal Oxide Semiconductor Field Effect Transistor. All three have several subtypes, and unlike passive semiconductor devices such as diodes, active semiconductor devices allow a greater degree of control over their functioning.
Depending on their subtypes, operating frequency, current, voltage and power ratings, all the three types of transistors come in a large variety of packages, and all of them are susceptible to ESD or Electro Static Discharge. That means when you handle these devices, you must take adequate precaution against static charges destroying them.
The basic construction of a BJT is two PN junctions producing three terminals. Depending on the type of junctions, the BJT can be a PNP type or an NPN type. The three terminals are identified as the Emitter or E, the Base or B and the Collector or C. BJTs usually function as current controlling switches. The three terminals can be connected in three types of connections within an electronic circuit – Common Base configuration, Common Emitter configuration and Common Collector configurations. All the three connections have their own functions, merits and demerits. The BJT is Bipolar because the transistor operates with both types of charge carriers, Holes and Electrons.
The FET construction does not have a PN junction in its main current carrying path, which can be made from an N-type or a P-type semiconductor material with high resistivity. A PN junction is formed on the main current carrying path, also called the channel, and this can be made of either a P-type or an N-type material. The three leads of a FET are the Source (S), Drain (D) and Gate (G), with Source and Drain forming the ends of the channel and the Gate controlling the channel conductivity. Unlike the BJT, the FET is a unipolar device since it functions with the conduction of electrons alone for the N-channel type or on holes alone for a P-channel type.
The input impedance at the gate of an FET is very high, unlike the BJT, which comparatively has much lower impedance. Additionally, the conductivity of the channel depends on the voltage applied to the Gate, essentially making it a voltage-controlled device, unlike the BJT, which is current-controlled. The voltage applied to the Gate controls the width of the channel, allowing the FET to carry current between the Drain and Source pins. The Gate voltage that cuts off the current flow between Drain and Source is called the pinch off voltage and is an important parameter.
The MOSFET is a special type of FET whose Gate is insulated from the main current carrying channel. It is also called the IGFET or the Insulated Gate Field Effect Transistor. A very thin layer of silicon dioxide or similar separates the Gate electrode and this can be thought of as a capacitor. The insulation makes the input impedance of the MOSFET even higher than that of a FET. The working of the MOSFET is very similar to the FET.
You can read more about transistors in depth here.