Applying and separating refer to whether two planets are moving toward an exact aspect (applying) or away from being exact (separating). An aspect is exact when the orb is zero. When the aspect is applying, the orb is getting smaller. When the aspect is separating, the orb is getting larger.
The planets must still be within orb of an aspect, but the strength of the aspect is often greater when the planets are applying than when they are separating.
Applying or separating only refer to how close in orb the planets are. Note that applying and separating are totally different from waxing and waning. It has nothing to do with where the planets are in the order of the zodiac signs – only whether they are moving toward the perfection of the aspect or away from it.
An aspect between two planets or points is Applying when (1) they are within orb of an aspect, and (2) they are getting closer toward being exact (i.e. the orb is decreasing). The effect of an applying aspect is usually considered stronger than a separating aspect, since the energy is still building toward a peak (exact).
An aspect between two planets or points is Separating when (1) they are within orb of an aspect, and (2) they are moving farther away from being exact (i.e. the orb is increasing). The effect of a separating aspect is usually considered weaker than an applying aspect, since the energy has already peaked (exact) and now its energy is dissipating.
Say the Moon is at 5° Virgo and the Sun is at 10° Taurus. They are 5° away from an exact trine aspect, which will perfect when the Moon reaches 10°. Astrologers may use very different orbs for aspects, but a trine's orb is usually somewhere around 5° to 6½° degrees. So, while the Moon is within orb to be trine the Sun, and before it reaches exact, the Moon is applying to the trine aspect with the Sun. When the Moon in Virgo reaches the same degree, minute and second as the Sun in Taurus, the aspect is exact.
The Moon then continues to move forward past 10°. They are still within orb of a trine aspect to each other, but now the aspect is separating. In other words, the orb grows larger; away from being exact.
Note that even though the Moon is farther ahead of the Sun in the zodiac, and the Moon is moving away from the Sun, it is still applying in terms of the trine aspect while the orb is growing smaller.
Usually an applying aspect will eventually perfect, and after it reaches exact it is then separating. However, occasionally one of the planets will change speed and the aspect starts to separate before it reaches exact. Kind of like a "near miss." This sometimes happens if one of the planets slows down to go retrograde or turn direct (usually the faster one).
An example of this is Mercury applying to a conjunction with Venus. Within about 8 degrees of each other, they are considered conjunct. As they approach the exact conjunction, the number of degrees decreases (the orb gets smaller). Since Mercury is generally faster than Venus, they will usually reach the exact conjunction within a few days.
However, if Mercury is about to go retrograde, its speed will start to decrease. At some point, Mercury will begin to go slower than Venus, which is about when their conjunction is no longer applying but is now separating. Eventually Mercury will slip away and will no longer be within orb of conjunction to Venus; Venus moves on without Mercury.
While they are within orb of being conjunct, their energies do combine but may fail to reach fruition. One way to interpret this is that you could get some creative ideas (Mercury-Venus) during this time, but you're not in a position to act on it or make it happen. After Mercury finally turns direct and gradually catches up to Venus again, you might be in a better position to act on the ideas from the previous conjunction period.
Another example of the "near miss" situation is when a planet goes retrograde just shy of making an exact aspect to another planet but still within orb—or when it retrogrades and stations direct just a few degrees away from an exact aspect to a planet. Some question whether the effects of a non-perfecting aspect like this could be insignificant; however the intensified stationary energy can reach across the gap to awaken the combined energies.