Linear relationships in the real world are pretty tricky. You almost always never have a truly linear relationship! But, usually, there are "linear regions" of these relationships that allow you to really simplify what's going on. Of course, too, when we talk about these slopes, we're assuming all other factors are held constant. So, let's talk about the slope.
If we're looking at a linear relationship with a positive slope, we're looking for two factors that correlate together. The easiest example for us would likely be the price-demand curve (demand line in basic macro). For example, if consumer demand doubles for Big Wheels, so will the price, more or less. As demand increases, price increases linearly.
If we want a negative slope, you could likely see easily that we could just flip from the price-demand curve to the price-supply curve. These two factors are inversely proportional, because if you add to one, you take away from the other. For example, if the de Beers company were to double the supply of diamonds in the world, the price of diamonds would drop proportionally. This is why they, and a couple of other major mining companies, control supply so tightly (near-constant demand + low supply = good profits)!
Now, let's look at some extreme cases that are harder to evaluate without deeper assumptions.
In the zero-slope case, we're looking at something where there is no change as the result of another variable's change. The easiest case would be when you're considering two completely unrelated variables. More interesting cases are found when we look at price ceilings and price floors. In a price-ceiling situation, a company may charge no more than, say, $2.00 for a banana because of government regulation. No matter how many more millions of people come clamoring for bananas (demonstrating increased demand) the company may only charge $2.00.
The infinite slope case is the opposite, where a variable CANNOT change. It could actually just be a rotation of the zero slope case, but there are much more interesting examples. Here the best example would be a pure staple product (demand is totally inelastic), where no matter what the price of the product is, people will never change their demand for it. This is easily seen with products like gasoline, electricity, or water. Sure, there is certainly some variation in demand in the real world, but if we consider a theoretical staple, there would be no change in demand.
I hope that helps!