The NOT gate is also called an "inverter", as its output is always the complement of the input. That is, if the input is TRUE then the output is FALSE else the output is TRUE end if). Its implementation electronically is not much different than that of an analog inverting amplifier, except that the gate is deliberately designed to operate nonlinear and change rapidly from one state to another.
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A "Not" gate is also called a "hex inverter". It's a binary logic gate that inverts the input. If the input is "1", the output will be "0".
An AND gate is a logic gate performing a Boolean logic AND operation.
The NAND gate has two or more inputs, and one output. This output is the complement of the AND of all the bits and will only be 0 if all the inputs are 1.A NOT gate on the other hand has only 1 input, and the output is the complement of this input.So to make a NAND gate into a NOT gate, we should tie (short, connect to same value) all the inputs of the NAND gate. At the output we would have the complement of the signal given at the tied inputs.This way we have a NOT gate from a NAND gate.
Fan-in is the max number of inputs a gate can take. Consequently, its importance depends on the type of application. A single gate with large fan-in can realize a logic function with many input variables as opposed to the other kind which might require more than one which causes its own problems like 0 or 1 hazards and increased gate delays..
An OR gate is a digital logic gate. If one or both of the inputs to the gate are "1", then the ouput of the gate will be "1" . If both of the inputs to the gate are "0" then the output of the gate will be "0".
A self-dual logic function is a function that is identical to its dual