The question could have been written better. I am assuming that you have two inputs each denoted by "a" and want to know which logic function requires both "a"s to be (1 or TRUE) so that the output is 1. The logic function is an AND gate
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Boolean is a 'true or false' logic in programming - if you define a function as a Boolean function, the only inputs it can have are true or false, and the output will vary dependant on the input
The acronym NAND is the short version of "Not AND". In terms of logic gates, a NAND gate refers to an output that is false only when all of its inputs are true.
a method of classification
multiplexer
Boolean algebra is used in logic circuits. Using And, Nor, Xor and Nand gates to determine the state of an output, dependant on the condition of various inputs. Or, if you like, to make a fixed decision based on the inputs. When designing a logic circuit, it is easy to get confused by too many steps in the process to get the answer that you want. By using Boolean algebra, the steps can be rationalised and reduce to the minimum number of steps, before committing to a finished physical circuit.