I am confused by what the buttons do on my Platinum ADC card (Penta, Trak Master, VoiceMaster Pro, TwinTrak Pro, TrakMaster Pro). What do I need to press in?

These buttons select the sample frequency. The button on the left swaps between 44.1 and 48 KHz. When the button on the right is pressed in it multiplies the sample frequency selected by two (hence the X2 marking) so now the button on the left swaps between 88.2 and 96KHz. As an example:

For 96KHz - left button out, right button in
For 48KHz - left button out, right button out