A linear system is a system that gives a predictable output based on superposition. What this means is if you put the sum of two signals into a system, you can expect the output to be a combination of the two outputs if the inputs were placed into the system by themselves.
So for example, lets say you put signal x into the system and the output is Ax. Then you put signal y into the system and the output is By. Then a linear system with signals x and y as input at the same time should have an output of Ax + By.
An example of a linear system would be a hot dog vendor. One day you put $3 into the system and and the output is one hot dog. Then the next day you put $6 into the system and get two hot dogs out. Then the third day, you bring your girlfriend and put $9 into the system (because you're paying for her lunch too) and get three hot dogs out.
Mathematically if the input is x[n] and output is y[n] (this is discrete time because no one would let you buy half a hotdog), then the relationship between the two are:
x[n] = y[n] (where x[n] represents 3 dollars input, y[n] represents 1 hotdog being output)
x1[n]-> y1[n]
$3 -> 1 hotdog
x2[n]-> y2[n]
$6 -> 2 hotdogs
x3[n] = x1[n] + x2[n]
$9 = $3 + $6
y3[n] = x3[n]
3 hotdogs = $9
y3[n] = y1[n] + y2[n]
3 hotdogs = 1 hotdog + 2 hotdogs
An example of a non-linear system would be if the hot dog vendor was having a sale of buy 3, get the 4th free. Like the previous example you would get 1 hot dog for $3, 2 hot dogs for $6, but you would get 4 hot dogs for $9.
x[n] = y[n] (where x[n] represents 3 dollars input, y[n] represents 1 hotdog being output)
x1[n]-> y1[n]
$3 -> 1 hotdog
x2[n]-> y2[n]
$6 -> 2 hotdogs
x3[n] = x1[n] + x2[n]
$9 = $3 + $6
y3[n] = x3[n]
4 hotdogs = $9
y3[n] = y1[n] + y2[n]
4 hotdogs = 1 hotdog + 2 hotdogs
