Filter order and cut off frequency
This answer is written in terms of low-pass but  the ideas directly translate to high pass. 

A butterworth filter (low pass) of order $N$ with cutoff $\Omega_c$ is one which has a maximally flat passband, and has the squared magnitude response $|H(j \Omega)|^2 = \frac{1}{1+\left(\frac{j \Omega}{j \Omega_{cutoff}}\right)^{2N}}$. [I'm just writing it this way so that if you try to get a realizable transfer function via spectral factorization it is easy]. 

So, you specify the cutoff $\Omega_c$ and it gets sharper as you increase the filter order. Strictly speaking, from the the squared magnitude response, you see that $|H(j \Omega)|^2=1$ if and only if $\Omega=0$, i.e. only the DC component is un-modified. However, from that formula, you can calculate things like frequencies where the magnitude is within say 99% of the original magnitude, by finding $\Omega$ such that $|H(j \Omega)|^2 \geq 0.99^2$ (note that $|H(j \Omega)|^2 $ is monotone decreasing in $\Omega$, so your range will be of the form $[-\Omega_{0.99},\Omega_{0.99}]$). Similar properties of other filters will hold as well - see Appendix B of Discrete-Time Signal Processing 2e by Oppenheim, Schafer and Buck for more details. Make sure you're familiar with the basic types of filters as well -- elliptical, butterworth and chebyshev to make sure your characteristics you desire are matched to what you want. 

If you want to design filters with particular passband and stopband characteristics in the FIR case (such as controlling the maximum error), you may want to look at the Parks-Mclellan algorithm (described in Discrete-Time Signal Processing 2e by Oppenheim Schafer and Buck, ch. 7).

As for Matlab usage, you can plot the magnitude response of the filter and determine these things. It can be done from the filter visualization tool. TI also makes a filter design tool which is quite nice, called FilterPro.