Meaning of 'number of parameters' in AIC When computing AIC,
$AIC = 2k - 2 ln L$
k means 'number of parameters'.  But what counts as a parameter?  So for example in the model
$y = ax + b$
Are a and b always counted as parameters?  What if I don't care about the value of the intercept, can I ignore it or does it still count?
What if
$y = a f(c,x) + b$ 
where $f$ is a function of c and x, do I now count 3 parameters?
 A: 
For any statistical model, the AIC value is $\mathit{AIC} = 2k - 2\ln(L)$
  where k is the number of parameters in the model, and L is the maximized value of the likelihood function for the model.

(see here)
As you may see, $k$ represents the number of parameters estimated in each model. If you model includes an intercept (that is, if you compute a point estimate, variance and confidence interval for the intercept) then it counts as a parameter. On the other hand, if you are computing a model without an intercept, it does not count.
Remember that AIC does not only summarise goodness of fit but it also considers the complexity of the model. That's why $k$ exists, to penalise models with more parameters.
I don't feel knowledgeable enough to answer your second question, I'll leave it for another member of the community.
A: As mugen mentioned, $k$ represents the number of parameters estimated. In other words, it's the number of additional quantities you need to know in order to fully specify the model. In the simple linear regression model 
$$y=ax+b$$
you can estimate $a$, $b$, or both. Whichever quantities you don't estimate you must fix. There is no "ignoring" a parameter in the sense that you don't know it and don't care about it. The most common model that doesn't estimate both $a$ and $b$ is the no-intercept model, where we fix $b=0$. This will have 1 parameter. You could just as easily fix $a=2$ or $b=1$ if you have some reason to believe that it reflects reality. (Fine point: $\sigma$ is also a parameter in a simple linear regression, but since it's there in every model you can drop it without affecting comparisons of AIC.)
If your model is
$$y=af(c,x)+b$$
the number of parameters depends on whether you fix any of these values, and on the form of $f$. For example, if we want to estimate $a, b, c$ and know that $f(c,x)=x^c$, then when we write out the model we have
$$y=ax^c+b$$
with three unknown parameters. If, however, $f(c,x)=cx$, then we have the model
$$y=acx+b$$
which really only has two parameters: $ac$ and $b$.
It is crucial that $f(c,x)$ is a family of functions indexed by $c$. If all you know is that $f(c,x)$ is continuous and it depends on $c$ and $x$, then you're out of luck because there are uncountably many continuous functions.
A: First, to those who may not be familiar with AIC: the Akaike Information Criterion (AIC) is a simple metric designed to compare the "goodness" of models.
According to AIC, when trying to choose between two different models applying to the same input and response variables, i.e. models designed to solve the same problem, the model with the lower AIC is considered "better".
In the AIC formula, $k$ refers to the number of variables (input features, or columns) in the model.  The more complex the model is (more variables needed to get the estimate or prediction), the higher the AIC is. This ensures that among two models with the same predictive power or accuracy, the simpler model wins.  This is a form of Occam's razor.
So the simple answer to the last question is: if the c in $f(c, x)$ is a constant that doesn't change with the observations, then, it should not be included in $k$.
