There are several unintended and problematic problems that occur in the conversion of digital data to analog signal. One is called "jitter" and remains a difficulty to this day. Jitter is measured in picoseconds, or trillionths of a second...yes, trillionths! Jitter occurs "when the distance between digital information, in time, is incorrect" (from Secrets of Home Theater and High Fidelity). When an "off" bit is followed by an "on" bit, and the "off" bit is read for even a tad too long a period of time, it crowds on to the time that the "on" bit would be normally read and confuses the interpretation.
This is a simplification of a very complicated factor, but regardless it is the timing of the data stream that is involved where jitter is concerned.
Many audio engineers feel jitter is not really an audible problem and dismiss it as unimportant. However, most new machines are being designed with low jitter in mind. A very common technique to reduce jitter is to store the digital data from the CD in a memory buffer where it can be reclocked and fed to the DAC in precise sequence. Vibration can also adversely affect CD player performance by causing jitter. To help eliminate vibration, use a quality damped housing.
The other big issue is the fact that it's difficult to accurately portray a complex audio signal in just 44,100 samples of 16-bits per second. DVDs we use today for video use 192,000 samples per second and produce 24-bit words of data! This is dramatically more data being recorded than that used in CDs. As such, the quality of the data is substantially better, which translates into noticeably better sound. DVD-A and SACD have up to 24 bit words collected at 192kHz to begin with, and the sound quality is spectacular.
Its very common in todays CD players to use something called "upsampling." Because modern DACs can handle the higher sampling rates and longer word lengths, some players have a chip that "upsamples" the data to higher rates. The upsampler tries to "interpolate" what the samples would have been if the original music had been sampled at the higher rate. The results are often quite good, but of course, nothing is better than having the actual music samples. Regardless of the tools used to defeat jitter or lack of samples, sound quality is the ultimate basis for a final decision on whether to purchase a particular CD player.
If you can ask for the specification on this if it is available when considering your purchase, do so. Seek the player with the lowest jitter. Jitter rates of 30 picoseconds or less are found in top quality CD players. CD players with high jitter can sound harsh and also lose fine detail, including the upper natural harmonics.