The FIR direct form implements a filter with impulse response $h[n]$ according to the convolution expression: $$ \begin{align*} y[n] = \sum_{m=0}^{M-1} h[m] x[n-m] \; . \end{align*} $$ This implementation is shown below and requires $N M$ multiplications.
When the impulse response is symmetric, we can reduce the number of multiplications due to redundancies in the coefficients based on a modified form of convolution. For an even number of symmetric coefficients, $$ \begin{align*} y[n] &= \sum_{m=0}^{M/2-1} h[m] (x[n-m] + x[n-(M-m-1)]) \end{align*} $$ For an odd number of symmetric coefficients, $$ \begin{align*} y[n] &= h[(M-1)/2] \, x[n-(M-1)/2]) + \sum_{m=0}^{(M-3)/2} h[m] (x[n-m] + x[n-(M-m-1)]) \end{align*} $$
Note that the equation above assumes $h[n]$ has even symmetry. We can also define expressions for an oddly symmetric $h[n]$. This implementation is shown below. It requires $N M/2$ multiplications.
The FIR cascade form implements a filter with impulse response $h[n]$ according to the Z-transform expression: $$ \begin{align} Y(z) &= X(z) H(z) &= X(z) \sum_{m=0}^{M-1} h[m] z^{-m} \\ &= X(z) \prod_{m=1}^{M-1} H_{m}(z) \\ \end{align} $$ In this scenario, $H_{m}(z)$ is a single-zero system $$ H_m(z) =1-z_m z^{-1} \; . $$ We can also write $y[n]$ this in time as $$ \begin{align} y[n] &= x[n] * h_1[n] * h_2[n] * \cdots * h_{M-1}[n] \; . \end{align} $$ Hence, this form passes our input signal thru $M-1$ different single-zero systems. This implementation is shown below.
If the impulse response $h[n]$ is real, we know that the zeros exist as a pair of complex conjugates (unless the zero is at 0 or -1). Hence, we can express some combine two transfer functions into $$ \begin{align} H_m(z) &= (1-z_m^* z^{-1}) (1-z_m z^{-1}) \\ &= (1 - (z_m + z_m^*) z^{-1} + |z_m|^2 z^{-2}) \end{align} $$ Hence, this form reduces the number of systems we cascade thru.
A symmetric FIR filter is a linear phase filter. Hence, if $z_m$ is a zero of the system, then $z_m^{-1}$ is also a zero. Hence, we can express each transfer function as $$ \begin{align} H_m(z) &= a_0 + a_1 z^{-1} + a_2 z^{-2} + a_1 z^{-3} + a_0 z^{-4} \end{align} $$ and we can use a symmetric direct form implementation to represent this. This further reduces the number of cascaded systems and reduces the number of multiplications.
The FIR frequency sampling form is based on the following expression for represening a transfer function: $$ \begin{align*} H(z) = \frac{1-z^{-N}}{N} \sum_{k=0}^{N-1} H[k] \left( \frac{1}{1 - e^{j\frac{2\pi}{N}k} z^{-1}} \right) \end{align*} $$
According to this derivation, an FIR filter can be represented by sum of $K$ systems. The $k$th component of the sum contains one pole, defined by $$ \begin{align*} p_k = e^{-j \frac{2 \pi}{N} k} \; , \end{align*} $$ and $N$ zeros (i.e., the roots of $1-z^{-N}$), defined by $$ \begin{align*} z_n = e^{-j \frac{2 \pi}{N} n} \; . \end{align*} $$ Hence, the pole overlaps with one zero in each component of the sum, canelling out the zero.
In this setup, the filter is described by an FIR filter (providing the zeros) cascaded with a sum of $K$ IIR filters (providing the pole). Each pole corresponds to a different frequency in the DFT. An advantage of this approach is that if we want to ignore certain frequencies, we can remove their corresponding IIR filters. This can significantly reduce our computational cost.