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.
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.
