As the enzyme concentration increases the rate of the reaction increases linearly.

Why? Because there are more enzyme molecules available to catalyse the reaction.

At very high enzyme concentration the substrate concentration may become rate-limiting, so the rate stops increasing - and is shown graphically as the levelling out of enzyme activity. Enzymes however, are typically present in cells at low concentrations.

As the substrate concentration increases, the rate of reaction increases.

Why? Because there are more substrate molecules which collide with enzyme molecules, thus, more enzyme-substrate-complexes form and more enzyme dependent reactions take place.

Graphically, the rate of an enzyme-catalysed reaction shows slight curved dependence on substrate concentration.

What is Km and Vmax?

Km (or Michaelis Constant) is the substrate concentration at which half the amount of an enzymes active sites are occupied by its substrate.

A high Km means lots of substrate must be available in order to saturate the enzyme - meaning the enzyme has a low affinity for its substrate. 

A low Km means only a small amount of substrate is needed to saturate the enzyme - meaning the enzyme has a high affinity its substrate. 

The maximum rate of reaction at infinite substrate concentration is called is maximum velocity or Vmax.

Vmax and Km are useful for characterising enzyme affinity and thus enzyme activity (rates of reaction). For example, a good enzyme will have a high Vmax and a low Km Meaning the enzyme dependent reactions will reach maximum velocity (Vmax) (maximum rate of reaction) with low concentrations of substrate (low Km).At higher concentrations the enzyme molecules become saturated with substrate, so there are fewer free enzyme molecules, so adding more substrate doesn't make any real difference to the rate of reaction (however, it will increase the rate of E-S collisions).

Enzymes have varying affinities (the tendency for the enzyme to bind to its substrate).

So, Enzymes have varying affinities for their substrates, and the affinity for an enzyme to bind to its substrates can be described by an enzyme's Km