The equation of a straight line is the best example from basic algebra of something that is linear. And indeed the word "linear" and the word "line" are from the same root.
In a more general sense, though, something is linear if what comes out is in direct proportion to what goes in. For example, in engineering, an ideal electric motor is linear. If you double the voltage you apply to it, it will turn twice as fast.
Pouring water into a graduated cylinder is linear. Each milliliter of water added increases the height of the water by a fixed amount.
A spring is linear. With each additional centimeter you stretch it, the force of the spring's reaction increases by a fixed amount.
If you are paid by the hour, then the amount of your pay is a linear function of the number of hours that you work.
These examples are only linear within a range. Outside that range they become nonlinear. The motor becomes nonlinear when you apply so much voltage that you damage it. The graduated cylinder becomes nonlinear when it overflows, and the spring when it reaches the limit of its ability to stretch. The pay example becomes nonlinear when you get paid time-and-a-half for hours beyond 40 per week.
Some examples of things that are inherently nonlinear:
The area of a circle as a function of its radius. The difference between the area of a circle of radius 1 and radius 2 is 4pi - pi = 3pi. The difference between the area of a circle of radius 10 and radius 11 is 121pi - 100pi = 21pi. So we see that each additional unit we add to the radius does NOT increase the area by a fixed amount.
Another example is pouring water into a conically shaped cup (like the ones you find at some water coolers). The first few milliliters fill the part of the cup near the apex of the cone. So each milliliter causes a large increment in the height of the water. As the cup fills, you are filling wider and wider sections of the cup, so the incremental height from each milliliter decreases and is NOT a fixed amount.