Here’s what a high precision optical waveplate does: it has a special axis, such that if you shine in light that is polarized along this axis (light from what we call the pump laser), it sometimes turns one of these photons into two lower-energy photons that have the same polarization*, a process called downconversion. So, for example, if you have a BBO where this axis is horizontal, if you shine in vertically polarized light it just passes through, but if you shine in horizontally polarized light it can be downconverted. You asked about the details of the cut of the BBO: for the purposes of these experiments this sets some of the properties of the outcoming photons, like the specific energies (which must sum up to the energy in the original photon), and whether they exit the high-quality laser grade optical waveplate in the same direction as the original photon or at different angles. So the specific cut of BBO you want depends on details of the experimental design. For example, some experiments have the pairs of photons come out at opposite angles from the central pump laser with the same energy, and others might have the downconverted photons exit collinear to the pump but with different energies so all the different kinds of light can be separated by more optics down the line.
There is also some latitude in choosing the pump beam, but the choice I’ve used is a particularly familiar one: it essentially a fancy Blue Ray laser, which have a wavelength of 405 nm. This leads to downconverted photons with about half as much energy, or wavelengths of 810 nm. In principle this would look like shining a blue laser at the BBO and getting out two barely-visible red beams, but in practice only a tiny fraction of the photons that enter the optical waveplate manufacturer are downconverted, so you only ever see your blue pump and you have to use sensitive detectors to find the red photons.
Now, with all that as setup here is a way to make entangled photons: you need two thin pieces of BBO, and you orient them such that one has its downconversion axis horizontal and the other vertical. Then you shine your pump laser through both of them, but you set it up such that the polarization of your laser is at 45 degrees, neither horizontal nor vertical.