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This might be a trivial question, but it seems to me that certain unvoiced example pronunciations for IPA sounds resemble the voiced one. For example, on this site:

http://www.internationalphoneticalphabet.org/ipa-sounds/ipa-chart-with-sounds/

The bilabial plosives p and b seem indistinguishable for me—they both sound like a voiced consonant; however, other pairs such as s and z are clearly distinguishable for me—s is unvoiced and z is voiced.

Is it just me who somehow can't tell the difference between the p b and t d pair? Perhaps it's just a characteristic of plosive consonants that the unvoiced sounds like a voiced consonant?

Typically, when a person cannot hear a difference between a voiceless stop and a voiced stop, as pronounced according to IPA principles, that is because the voiceless stop is unaspirated, and the person listening is a speaker of English. In English, syllable-initial pre-stress /p,t,k/ are aspirated. Furthermore, voiced stops in that position are not fully voiced and for many speakers are actually unvoiced. So a phonetic transcription of "pin" would be [pʰɪn], and "bin" would be [pɪn]. The consonants in those performances are not pronounced as English consonants.

If you try to hear those sounds "conceptually", that is by categorizing the sound as "p" or "b", and your reference point is English, what you are hearing is not English "p" or "b", and they both sound close enough to "b" that you think they are the same, namely "b". An alternative approach would be to compare that IPA performance of p vs. b and ask, do these really sound exactly alike? I would also suggest using this chart or this chart, which are performed by acknowledged phonetic experts. If you absolutely can't hear any difference in [ata] and [ada], that's an interesting problem – usually people can hear that they are different, the problem is just how to classify what you hear.

Speakers of other languages have different experiences. For example, speakers of Hindi typically have no problem getting the distinction since their language has unaspirated and fully voiced stops.

Many native English speakers have the same reaction as you, because they use the fortis/lenis (strong/weak) articulatory difference to distinguish [pa]/[ba], not the unvoiced/voiced difference. This leaves them free to devoice the [b], which in fact they often do, since it is less effort. So, it is likely that your pronunciation of "bah" has a devoiced "b", which makes it [pa], really.

The reason /b/ tends to be phonetic [p] is that there is no easy way to maintain the voicing of [b], because the air that must continue to escape through the glottis has no way to get out of your mouth. One thing you can do, which I refer to as the Bing Crosby maneuver, is to loosen the muscles of your cheeks to let them puff out, so the oral cavity has room to contain the extra air for the brief time the [b] needs it. Otherwise, the increasing air pressure in the mouth will choke off the voicing in the glottis by slowing the flow of air there. Dropping the jaw or the larynx has a similar effect.