Hagen von Eitzen answered the question perfectly assuming that the system you are looking at is discrete time (i.e., defined by the difference equation $x(n+1) = f(x(n))$). However your comments in the question (e.g., if $a=0$ there is only one fixed point) make me suspect you are looking at a continuous time system (i.e., defined by the ODE $\dot{x}(t) = f(x(t))$. This said, the fact that you added tag chaos-theory might imply that you are actually looking at the discrete time case (my point is that making sure your question is unambiguous helps a lot for someone trying to answer it). Anyway, for completeness I post the continuous time answer as well.
A fixed point of $\dot{x}=ax-x^3$ is a real number that solves $\dot{x} = 0 = ax - x^3 = x(a-x^2)$. Thus $x_{fp}=0$ is always a solution irrelevant of the value of $a$. In addition, if $a>0$ we get two extra solutions $x_{fp}=\pm\sqrt{a}$. We have found the fixed points, next we can linearise around the fixed points to study how they behave locally. The Hartman-Grobman Theorem says that a fixed point $x_{fp}$ locally (in a small neighbourhood of $x_{fp}$) behaves as an asymptotically stable (unstable) fixed point if $\delta_x = 0$ is an asymptotically stable (unstable) fixed point of the `linearised system'
$\dot{\delta}_x = \frac{df(x_{fp})}{dx}\delta_x$.
The linearised system is asymptotically stable if and only if $\frac{df(x_{fp})}{dx}<0$ and unstable if and only if $\frac{df(x_{fp})}{dx}>0$. Notice that the Hartman-Grobman Theorem is not applicable if $\frac{df(x_{fp})}{dx}=0$. Then
$\frac{df(x)}{dx} = a-3x^2$
thus if $a<0$, $\frac{df(0)}{dx} = a <0$ and the unique fixed point $x_{fp}=0$ behaves locally as an asymptotically stable fixed point. If $a=0$, $\frac{df(x)}{dx} =0$ and so we get no information by linearising. Lastly, if $a>0$ then $\frac{df(0)}{dx} = a>0$ and $\frac{df(\pm\sqrt{a})}{dx} = -2a<0$, so $x_{fp}=0$ is unstable and $x_{fp}=\pm\sqrt{a}$ behave locally as an asymptotically stable fixed points.
Just a side note, linearising systems of order one is often a bit pointless given that you can draw conclusions regarding the global behaviour of the system by simply examining the phase line (continuous time) or cob-webbing (discrete time). For more info and a nice introductory book on non-linear systems see
http://www.amazon.co.uk/Nonlinear-Dynamics-Chaos-Applications-nonlinearity/dp/0738204536.