In the context of a bass, the term hardware describes parts like the bridge, tuning pegs, strap pins and the controls for the electronics.
The most important piece of hardware is without doubt the bridge, as it carries the vibrations of the strings into the body of the instrument. It is obvious that by choosing different materials and thicknesses, different sonic characteristics can be achieved.
Some especially convenient bridges let you drop in the strings from the top (these are called top loader bridges). On other models, the strings must be fed through holes and sometimes even through the body ("string through body"). While this makes changing the strings more tedious, some bassists are of the opinion that string-through-body instruments resonate better.
An important measure of quality for a bridge is its ease of use when adjusting the string action of the instrument. Can the position of the string be adjusted two- or even three-dimensionally? The more flexible the bridge, the easier it is to achieve an ideal string action that suits your personal preference.
The rigidity and free movement of the tuning mechanisms on the head of the bass is another criteria of quality. That said, cheap tuners can be replaced by better models relatively easily.
Hint: The so-called D-tuner is a small lever sold by the American company Hipshot, which allows for quick adjustment of the low E string down to D even during playing. While this is only a small adjustment, it can expand your sonic possibilities drastically, especially in harder styles. D-tuners are sold in many varieties for different models of basses, and you can fit one to your instrument yourself in a few easy steps.