On the more fundamental level of ear training, the ability to hear how "in tune" an instrument is. Though seemingly basic, the skill can vary greatly in complexity. In one instance, a musician can detect if a certain instrument is above or below the another's pitch (similar to how a singer that is off key can be pointed out) but a truly advanced ear can identify without another instrument as as reference if an instrument is set into a different tuning arrangement (baroque vs contemporary).
Someone with true perfect pitch can identify a note's name instantly and accurately. There shouldn't be too much thinking involved in the process (alternately, some can think about the note for a few moments before coming up with the correct answer--this is a skill known as relative pitch as the musician is identifying the pitch relative to a known pitch). Practicing with a keyboard is the only true way to gain this skill. At first it is simple trial and error. However, just listening actively to music for a prolonged period can sharpen this skill.
Chords, which are essentially, collections of notes played simultaneously, are the basis for melodic direction. The attuned ear can hear the chords in music quite clearly and is able to write them out as they are occurring.
Composers will often change the key of their pieces to heighten the excitement and further the complexity of the melody. Although some key modulations are very noticeable, think of Gospel songs where the key is taken up a half step near the end, others are more subtle. A person with perfect pitch can notice the minutia of all key changes and be able to identify the original key and where it has changed to.
Rhythms are the pulse of the music and are equally as important as the pitches. In training a musician's ear, it is important to be able to identify what the rhythms and tempos are. On a basic level, a trained ear should be able to write out the rhythms that an instrument is playing with precision.