Parks-McClellan:

The Parks-McClellan method (called "Remez" by Matlab) is probably the most widely used FIR filter design method. It is an iteration algorithm that accepts filter specifications in terms of passband and stopband frequencies, passband ripple, and stopband attenuation. The fact that you can directly specify all the important filter parameters is what makes this method so popular. The PM method can design not only FIR "filters" but also FIR "differentiators" and FIR "Hilbert transformers".

In the windowing method, an initial impulse response is derived by taking the Inverse Discrete Fourier Transform (IDFT) of the desired frequency response. Then, the impulse response is refined by applying a data window to it.

The impulse responses of certain types of FIR filters (e.g. Raised Cosine and Windowed Sinc) can be calculated directly from formulas.