We have lots of common everyday experience with positive pressure, the canonical example is a gas.
But other examples of positive pressure are easy to imagine: for instance, a solid that gets compressed to be more compacted than its equilibrium density.
To me it is straightforward that if a solid is instead pulled apart slightly so that it is still connected but at a lower density than its equilibrium density, that it can have a tension that is a negative pressure.
But sometimes people object to negative pressure, so I think we could benefit for a comprehensive answer, that includes good definitions, justifications about why the definitions are good, and even includes comparisons to energy conditions (weak energy condition, strong energy condition, dominant energy condition, etcetera).
An answer does not need to address cosmological constants or dark energy specifically, but I would like the answer to be comprehensive enough that people with questions about those issues can satisfy all their questions about negative pressure itself.
What is negative pressure in general? How do we know that is the proper and fully general definition? Is it reasonable in light of known and acceptable physics? How/why do we know that? How, if at all, does it relate to tension? If different than tension, what is tension in general? How do we know that is the proper and fully general definition? Is it reasonable in light of known and acceptable physics? How does negative pressure relate to the classical energy conditions? Are any deviations or clashes with classical energy conditions justifiable or acceptable?