While in animals with complex nervous systems like humans and also many mammals and birds there may be psychological reasons for suffering, like the absence or death of someone beloved, suffering from physical pain is present in most, if not all animals.
The sensation of pain is provided by dedicated sensory neurons, like other sensory neurons are specialized for sensing light, sound, smell, taste, temperature, tactile pressure, gravity, force in the muscles/tendons, electric currents, magnetic fields, radiant heat a.k.a. infrared light and so on (some of these sensors exist only in some non-human animals).
The pain-sensing neurons, a.k.a. nociceptors, can be identified anatomically in some of the better studied animals, including humans, but it is likely that they also exist in most other animals, with the possible exception of some parasitic or sedentary animals, where all the sense organs are strongly reduced.
So all animals with such sensory neurons that cause pain are certain to suffer.
The nociceptors are activated by various stimuli, e.g. either by otherwise normal stimuli that exceed some pain threshold, e.g. too intense light or noise, or by substances generated by damaged cells from their neighborhood.