Q:

Is an atmosphere with different layers possible?

I'm making a world where you live in the air, as the surface is not survivable (like Venus). I'm going to leave out the story of the planet, as it's not important.
Anyway, could there be a planet with different layers in its atmosphere? For example, the bottom layer, closer to the surface would be dense, heavy gas (or a mixture of gases), while higher up, there is a layer of lighter gas (or a mixture of gases), something like Earth's nitrogen and oxygen breathable atmosphere.
Is this even possible, and if so, what would the transition between the layers be like (smooth and gradual, sudden, etc.)?
Also, for a bonus question, what would the most likely gases in these layers be?

A:

Yes
But,
there are many things working against this forming - especially for a terrestrial world.  Here are the issues that spring to mind:
Chemistry
You can subdivide chemicals using many different strategies.  I'm going to subdivide them by calling them:  

reducers (fuels)
oxidizers
inert

If you have an atmosphere with a high quantity of oxidizers (like on Earth), then reducing gases (like methane) added to the atmosphere don't last long.  They react quickly and are flushed out.
So really your atmosphere can only contain inert (like $N_2$) & oxidizers, like Earth or inert (like $He$) and reducers (like $H_2$ & $CH_4$), like Jupiter.
Fluid Dynamics
Fluids (liquids and gases are both considered fluids) tend to mix.  There are two primary mechanisms for this:  

Diffusion
Mechanical mixing (usually by turbulent boundaries between the layers)

The primary gases of the Earth's atmosphere are:
$$\begin{array}{|c|c|c|c|}
\hline \text{Gas} & \text{Fraction} & \text{Atomic Mass} & \text{Type} \\
\hline \text{Nitrogen } N_2 & \text{78%} & \text{28} & \text{Inert} \\
\hline \text{Oxygen } O_2 & \text{21%} & \text{32} & \text{Oxidizer} \\
\hline \text{Argon } Ar & \text{1%} & \text{40} & \text{Inert} \\
\hline \text{Carbon Dioxide } CO_2 & \text{<<1%} & \text{44} & \text{Inert} \\
\hline \text{Water } H_2O & \text{Up to 1%} & \text{18} & \text{Inert} \\
\hline
\end{array}$$
Note the absence of compounds considered reducers.
Despite the wide range of atomic masses, the Earth's atmosphere is generally well-mixed.  This is due to wind.  In some locations with volcanic venting you can get pools of (invisible) unmixed $CO_2$ gas.  Animals (including humans) that wonder into these pools die from asphyxiation.

Carbon dioxide, being about 1.5 times as dense as air, caused the
  cloud to "hug" the ground and descend down the valleys, where various
  villages were located. The mass was about 50 metres (160 ft) thick and
  it travelled downward at a rate of 20–50 kilometres per hour (12–31
  mph). For roughly 23 kilometres (14 mi) the cloud remained condensed
  and dangerous, suffocating many of the people sleeping in Nyos, Kam,
  Cha, and Subum.

List of candidate gases
From lightest to heaviest (in my list).  The greater the difference in the gases masses, the more likely you'll get differentiation.
$$\begin{array}{|c|c|c|}
\hline \text{Gas} & \text{Atomic Mass} & \text{Type} \\
\hline \text{Hydrogen } H_2 & \text{2} & \text{Reducer} \\
\hline \text{Helium } He & \text{4} & \text{Inert} \\
\hline \text{Methane } CH_4 & \text{16} & \text{Reducer} \\
\hline \text{Ammonia } NH_3 & \text{17} & \text{Reducer} \\
\hline \text{Water } H_2O & \text{18} & \text{Inert} \\
\hline \text{Neon } Ne & \text{20} & \text{Inert} \\
\hline \text{Nitrogen } N_2 & \text{28} & \text{Inert} \\
\hline \text{Carbon Monoxide } CO & \text{28} & \text{Reducer} \\
\hline \text{Nitric Oxide } NO & \text{30} & \text{Reducer?} \\
\hline \text{Oxygen } O_2 & \text{32} & \text{Oxidizer} \\
\hline \text{Hydrogen Sulfide } H_2S & \text{34} & \text{Reducer} \\
\hline \text{Fluorine } F_2 & \text{38} & \text{Oxidizer} \\
\hline \text{Argon } Ar & \text{40} & \text{Inert} \\
\hline \text{Carbon Dioxide } CO_2 & \text{44} & \text{Inert} \\
\hline \text{Nitrous Oxide } N_2O & \text{44} & \text{Oxidizer} \\
\hline \text{Nitrogen Dioxide } NO_2 & \text{46} & \text{Oxidizer} \\
\hline \text{Sulfur Dioxide } SO_2 & \text{64} & \text{Reducer} \\
\hline \text{Chlorine } Cl_2 & \text{70} & \text{Oxidizer} \\
\hline \text{Krypton } Kr & \text{84} & \text{Inert} \\
\hline \text{Xenon } Xe & \text{131} & \text{Inert} \\
\hline \text{Bromine } Br_2 & \text{160} & \text{Oxidizer} \\
\hline \text{Iodine } I_2 & \text{254} & \text{Oxidizer} \\
\hline
\end{array}$$
Light gases are dominated by reducing chemicals while heavy gases are dominated by the oxidizing chemicals (primarily the Halogens).  Inert gases like $N_2$ and the noble gases are sprinkled through the mix.
One Possible Setup
As long as the reducing layers and oxidizing layers are separated by inert layers, you might be able to have all sorts of gases.
On possible configuration would be (with the Bromine layer as the deepest layer).
$$\begin{array}{|c|c|c|}
\hline \text{Layer} & \text{Gas} & \text{Type} \\
\hline 1 & \text{Bromine} & \text{Oxidizer} \\
\hline 2 & \text{Krypton} & \text{Inert} \\
\hline 3 & \text{Earth's Atmosphere} & \text{Oxidzer} \\
\hline 4 & \text{Helium} & \text{Inert} \\
\hline 5 & \text{Hydrogen} & \text{Reducer} \\
\hline
\end{array}$$

