Why do logic gates behave the way they do?
As stated by user120366, 16 possible 2-input logic gates exist, I've tabulated them for you here:

A|B||0|1|2|3|4|5|6|7|8|9|a|b|c|d|e|f
-+-++-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-
0|0||0|0|0|0|0|0|0|0|1|1|1|1|1|1|1|1
0|1||0|0|0|0|1|1|1|1|0|0|0|0|1|1|1|1
1|0||0|0|1|1|0|0|1|1|0|0|1|1|0|0|1|1
1|1||0|1|0|1|0|1|0|1|0|1|0|1|0|1|0|1


A and B are the inputs, 0 through f are the possible permutations of outputs.

These gates have been named:

0 = FALSE 
1 = AND
2 = A NIMPLY B (A AND NOT B)
3 = A
4 = B NIMPLY A (B AND NOT A)
5 = B
6 = XOR
7 = OR
8 = NOR
9 = XNOR
a = NOT B
b = B IMPLY A (A OR NOT B)
c = NOT A
d = A IMPLY B (B OR NOT A)
e = NAND
f = TRUE


6 of these (0,3,5,a,c,f) discard one or both inputs. The IMPLY and NIMPLY gates are rare, though they are certainly used in formal logic. AND, OR and XOR are easiest for humans to reason with, but for physical hardware, NOR and NAND are also used heavily, because they can be simpler to implement and make smaller circuits. The same probably holds for XNOR.

So, as stated earlier, it's not so much that we decided that the gates should behave this way, but that 16 possible gates can be defined, and we came up with descriptive names for them.