How does Naive Bayes classifier work for continuous variables?
The difference boils down to "how we define $P(x_i|C_k)$?", where $x_i$ is a single feature, and $C_k$ is a class from a total of $K$ classes. 

Discrete

In discrete case,  $P(x_i|C_k)$ is represented by a table as follows:

x_i   P(x_i|C_k)
a     0.5
b     0.2
c     0.3


We have one of these tables for each feature-class pair $(i, k)$. 

Lets denote $i$-th feature of data point $n$ as $x_{n,i}$. Each row of this table can be estimated using 

$$\hat{P}(x_i=a|C_k) = \frac{\sum_{n:n \in C_k}{\mathbb{1}_{x_{n,i}=a}}}{N_k}$$

which divides the number of samples that have i-th feature equal to $a$ by total number of samples in class $C_k$ (of course from the training set).  

Also, check out pseudocount that avoids zero estimations.

Continuous

In continuous case, we either discretize the continuous interval of $x_i$ into bins $\{b_1,..,b_m\}$ and proceed the same as discrete case, or we assume a function like Gaussian (or any other one), as follows:

$$P(x_i|C_k)=\frac{1}{\sqrt{2\pi}\sigma_{i,k}}e^{-(x_i-\mu_{i,k})^2/2\sigma_{i,k}^2}$$

This way, for each feature-class pair $(i, k)$, $P(x_i|C_k)$ is represented with two parameters $\{\mu_{i,k}, \sigma_{i, k}\}$ instead of a table in discrete case. The estimation of the parameters is the same as fitting a Gaussian distribution to one dimensional data, that is:

$$\hat{\mu}_{i,k} = \frac{\sum_{n:n \in C_k}{x_{n,i}}}{N_k}, \hat{\sigma}^2_{i, k} = \frac{\sum_{n:n \in C_k}{(x_{n,i} - \hat{\mu}_{i,k})^2}}{N_k-1}$$

Instead of Gaussian, we can opt for a more complex function, even a neural network. In that case, we should look for a technique to fit the function to data just like what we did with Gaussian.

The rest is independent of feature type

Representing $P(C_k)$ is the same for both discrete and continuous cases and is estimated as $\hat{P}(C_k)=N_k/N$.

Finally, the classifier is $$C(x_n) = \underset{k \in \{1,..,K\}}{\mbox{argmax }}\hat{P}(C_k)\prod_{i}\hat{P}(x_i=x_{n,i}|C_k)$$

Or equivalently using log-probabilities,

$$C(x_n) = \underset{k \in \{1,..,K\}}{\mbox{argmax }}\mbox{log}\hat{P}(C_k)+\sum_{i}\mbox{log}\hat{P}(x_i=x_{n,i}|C_k)$$