Aspects have two different factors that influence how they work. The first is the "whole sign" relationship (that is, how well do the two signs in question work with each other). The second is the harmonic vibration, which has to do with how close the two planets are to being in a specific relationship to each other that corresponds to the division of the 360 degree circle by a whole number.
There are two classifications of Aspects: "Major" and "Minor." The "Major" aspects (Conjunction, Opposition, Trine, Square, and Sextile) all share both strong harmonic and sign relationships. The Quincunx has only recently been "upgraded" to a major aspect. More than anything, what makes a "Major" aspect "Major" is how strong and noticeable it's ultimate effects are. The "Minor" aspects involve less obvious divisions of the chart, or harmonics that operate on other levels than the physical.
The Major Aspects
The Conjunction is a 1st harmonic aspect - the division of the circle by 1. Two planets are conjunct if they occupy the same position in the chart by sign and degree. When two planets are conjunct, they tend to blend their energies and always act together. Conjunctions are extremely important aspects, and the closer the conjunction is, the more important the aspect will be to the individual. Conjunctions also represent the beginning of a new cycle involving the two planets.
The Opposition is a 2nd harmonic aspect - the division of the circle by 2, resulting in a 180 degree aspect. Planets in opposition are directly across the chart from each other, and therefore can "see" each other very clearly. Oppositions are always about learning how to balance the energy and expression of the planets and signs involved. Sometimes when two planets are in opposition, it can feel like only one can be 'used' at a time; if one is "on" the other must then be "off". The key with oppositions is learning how to integrate the two planets so that they can both function at the same time.
The Trine is a 3rd harmonic aspect - the division of the circle by 3, resulting in a 120 degree aspect. The 3rd harmonic is very creative, although it represents more creative potential than it does creative expression. Planets in Trine are usually in the same element (Fire, Earth, Air or Water), and this represents a very harmonious flow of energy between the two planets. Trines represent inborn talents and gifts. However, Trines are notoriously lazy and require a conscious effort to make use of their gifts.
The Square is a 4th harmonic aspect - the division of the circle by 4, resulting in a 90 degree aspect. The 4th harmonic is where things first begin to manifest on the material plane. Planets that are square each other are usually in the same Modality (Cardinal, Fixed or Mutable) and squares can feel as if the planets are working at cross-purposes to each other. Squares create tension and conflict, and are generally uncomfortable. However, squares are also how we achieve things, and get things done.
The Sextile is a 6th harmonic aspect - the division of the circle by 6, resulting in a 60 degree aspect. The sixth harmonic is considered to be one of balance, harmony and rest. Planets that are Sextile each other are usually in the same polarity, but different elements (i.e. Earth and Water (both feminine) or Fire and Air (both masculine)). Sextiles represent an opportunity for the two planets to work with each other in a flowing and harmonious way; however, unlike Trines where the flow is always there, Sextiles must be activated to take advantage of them. Sextiles truly represent opportunities.
The Quincunx is actually a 12th harmonic aspect - it is 5/12 of the circle, or a 150 degree aspect. Signs that are Quincunx each other have nothing traditional in common - they share nothing by polarity, element or modality (although some of the signs connect in other, less obvious ways). When planets are Quincunx each other, they can often feel like an opposition in that there should be a way to balance them, and we may try to. However, there isn't a way of balancing these planets, and Quincunxes generally require some sort of adjustment.
The Minor Aspects
The Semi-Square is an 8th Harmonic Aspect - the division of the circle by 8, resulting in a 45 degree aspect (or half of a square). The 8th harmonic, like the 4th harmonic is also action oriented, and operates on the material plane. However, because of the 2nd harmonic aspect (2 x 4th harmonic), the 8th harmonic also introduces an element of balance and rest. Semi-Squares occur in signs that share a common polarity (masculine (Fire and Air) or feminine (Earth and Water)), and therefore there is the potential for harmony between them. This is still a "hard" aspect in that it can be uncomfortable and force us to take some action in order to resolve the discomfort; however, Semi-Squares can be resolved more quickly than squares because of the ability to find that point of balance.
The Sesquiquadrate is also an 8th Harmonic Aspect - it is 3/8 of the circle or 135 degrees. Sesquiquadrates tend to be slightly less action-oriented than the Semi-Squares. They still represent a feeling of tension, and a need for a release, of course. However, planets that are Sesquiquadrate each other, are usually 5 signs apart, and are therefore in signs that are Quincunx each other and have nothing in common. Sesquiquadrates are often best handled through humor - often being able to joke about the associated situations can release the tension.
The Semi-Sextile is a 12th harmonic aspect - it is 1/12 of the circle or 30 degrees. Planets that are Semi-Sextile to each other are in signs that are adjacent to each other. Like the Quincunx, the Semi-Sextile signs have nothing in common by either polarity, modality or element. However, unlike the Quincunx the Semi-Sextile is usually an easier aspect to work with because each sign naturally evolves to the next sign. The Semi-Sextile is usually considered to be a weakly harmonious influence, one that requires even more effort to activate and take advantage of than the Sextile.
The Quintile is a 5th harmonic aspect - it is 1/5 of the circle or 72 degrees. The Quintile is not an easy aspect to see directly, and this is one of the reasons that it is not often used by many astrologers. The 5th harmonic relates to creativity in action, and tends to operate on the mental plane rather than the physical plane. Quintiles have been linked to great creativity, and, being a 5th harmonic, also to a need for freedom and change.
The Septile is a 7th harmonic aspect - it is 1/7 of the circle, and the first harmonic that does not evenly divide the circle. Septiles are 51 1/2 degree aspects, and are extremely difficult to see in the chart without the aid of a computer program or an aspect dial. The 7th harmonic has been associated with matters of fate and karma, and also with spirituality and our connections to our higher self. In numerology, the number 7 is the first number that we can not grasp by just looking at it (we can tell how many objects there are up to six without having to count or break them down into smaller groups). The number 7 is just beyond our ability to comprehend. Personally, I feel that Septiles relate more to our spiritual lessons than anything else, and that they require a conscious effort in order to experience or work with them.
The Novile is a 9th harmonic aspect - it is 1/9 of the circle, or 40 degrees. In numerology, the number 9 relates to the end of a cycle; there is an element of completion and perfection here, as well as an element of being tested before being allowed to move on to the next cycle. Noviles (also called Noniles) have been linked with perfection and completion, and they have also been linked (somewhat less appropriately, I feel) with imprisonment and difficulty. Again, I feel that Noviles tend to operate on a more spiritual level than on a mundane level, and that they do indicate that we will be tested on our skills in working with the two planets involved; they also indicate that we have reached a point with these two planets where we have adequate resources and experience to pass the test - all we need to is integrate this and let ourselves be guided by spirit. In Jyotish (Vedic) astrology, the 9th harmonic chart is the chart of Marriage.