An aspect is a very useful part of Astrology. Johannes Kepler defines an aspect thus;
An aspect is an angle formed at the Earth of the beams of two Planets or stars, at a certain proportionate distance one from another, and have great power to stir up influence.
With this we can understand that an aspect is very influential in the judgment of the Planets. There are two main groups of aspects; Ptolemaic and Keplerian. As you may have already guessed, the first group was first written about by Ptolemy and therefore credited to his creation. These aspects are the conjunction, opposition, trine, square and sextile. These are the basis and main aspects created by the Planets and used in all branches of Astrology. The second type are Keplerian, these are aspects created by the famous Astrologer and Astronomer Johannes Kepler. They are however less used and less useful than the Ptolemaic aspects. These aspects included the semi-quadrate(or square), semi-sexile, quintile, biquintile, sesquiquadrate, & quincunx. The only useful ones primarily are the semi-quatrate and the semi-sextile, both of which are primarily (and almost exclusively) used and observed in medical Astrology. Since there is relatively little use for these aspects, I have simply described here for you the Ptolemeaic aspects.
Conjunction
A conjunction is technically not considered an aspect because unlike the other aspects, there is no set degree and therefore is not considered an aspect. A conjunction occurs when two Planets come into each others orb allowance and thus join together through body. When this happens the influences and significations of the Planets commix. In order to determine which Planet is most powerful in a conjunction (or any Aspect), look to which Planet has the most dignities in that placement and he will be the stronger one. A conjunction by itself is neutral because it is completely reliant on the two Planets who are forming the conjunction itself. An example of a conjunction is thus, Jupiter in 10° of Aries and Saturn 10° of Aries. The further each Planet is from 0° on eaither side, the weaker the influence each Planet has on each other.
Opposition
It is important to first note that the Aspects, as mentioned in the House dignity Treatise, follow the benevolent or malevolency degrees which are from the Sun. The opposition is a malevolent aspect because both Planets look directly at eachother and therefore, butt heads with each other; thus they are separated by 180°. This aspect is especially detrimental when it happens between the Sun and Moon because of the immense power they possess over life on earth. An example of Opposition would be Saturn at 10° Aries with Jupiter 10° Libra. Planets in opposition are divided by 6 Signs. The opposition is often called "perfect hate" or "perfect enmity" because it is so malevolent and malicious. This aspect is interpreted as the unfulfillment of a desire or a lot of work, with little reward. The benevolents in Opposition with each other are always less damaged by this aspect then the Malevolents and the Neutral Planets. As with the conjuntion (and all other aspects) the further each Planet is from 180° the weaker the aspect force is.
Trine
A Trine is a 120° application (or separation) of two Planets. In a Trine there are 5 signs between both Planets. A Trine is considered to be perfect love or perfect amity between two Planets because they receive each others rays harmoniously. The Planets are both far enough from each other not to see the flaws held by each of them but close enough to spread their benevolence to each other and thus they agree and get along. A Trine is the most beneficial aspect that can be found in any chart. It matters very little if the nature of the Planets are opposite because of their degree of separation they are quite content with one another. An example of a Trine is Saturn in 10° Aries with Jupiter 10° Leo, this is a Partile Sinister Trine. This aspect is interpreted as very little work with a large reward from the desired outcome.
Square
A Square aspect is when two Planets are applied (or separated) by 90°. This is the lesser malevolent aspect and thus termed imperfect hate or imperfect enmity. Two Planets in the aspect of a Square are divided by 4 signs. This aspect draws on suspicion of the Planet it views out of the corner of its "eye" but because it is not directly in the view of the Planet, its malevolence is slightly abated and thus the Square aspect is not as malevolent as the Opposition aspect. An example of a Square is Jupiter at 5° Aries, squared Saturn at 5° Cancer; this aspect is a Partile Sinister Square. This aspect is interpreted as frustrating tribulations with regards to desires but the outcome is generally in the favor of the Planet whom is strongest.
Sextile
A Sextile aspect happens when two Planets are within their orbs applying (or separating) each other by 60°. A Sextile divides the Planets by 3 signs. The Sextile is considered the aspect of imperfect love or friendship. It is less benevolent than the Trine because it is harder for the applying Planet to see the Planet it applies to however because of its closeness with the Planet it is considered by friendship. An example of a Planet in Sextile with another is if Jupiter was 10° Aries and Saturn was 10° Gemini. This aspect is interpreted as a slowness to gaining results desired but the results are obtained with little difficulty.
Sinister & Dexter
There are four major aspects and a faux-aspect known as Conjunction in Astrology. Three of these aspects (Trine, Sextile & Square) can be broken into two parts. These parts were called by the ancients as Sinister and Dexter. A Sinister aspect is one that is made with the succession of the signs and a Dexter aspect is one that is made with the anti-succession of the signs. For example, a Planet in Libra is in Sinister square aspect to a Planet in Capricorn because Libra to Capricorn follows the succession of the signs. A Planet in Libra making a square unto a Planet in Cancer is thus making a Dexter square aspect because it follows the anti-succession of the signs. This is easily found out by following the succession of the Houses, which also follow the succession of the signs.
Sinister and Dexter aspects are important to consider as they can add power to a specific aspect. It is better to have a Sinister aspect than a Dexter for obvious reasons that following the succession of the Signs brings and easier aspect than going contrariwise to the Signs. Conjunctions and Oppositions are not considered for obvious reasons. Included isa table to help the students who need a visual in order to grasp the concept.
Partile & Platic
Partile & Platic, as with Sinister & Dexter aspects are a state of an aspect and not aspects themselves. Both of these influence the power of the aspect. A Partile aspect happens when a Planet is in 17' (minutes) of application or separation of its conjoining Planet. This aspect is most prominently used when a Planet makes a Partile aspect to the Sun as the ancients have labeled it Cazimi. Before (or after) Planets are in Partile aspects, they are in what is called a Platic aspect. This is their regular orb allowance outside of 17'. Therefore Jupiter found at 10°00' and Mars at 9°13'A or 10°17'S it is considered in Partile aspect with Jupiter. If Mars were 9°12'A or 10°18'S he would be in Platic aspect of Jupiter because he is outside the orb allowance of 17' (applying or separating).
As with Sinister & Dexter, Partile aspects are more powerful and stronger and Platic are weaker. When two Planets are in Partile and there is a malevolent aspect between them, the malevolency is amplified more vehemently than it would be in a Platic aspect, the contrary being true if the aspect is a benevolent one and this concludes the treatise on the Aspects of the Planets.
Latitude
For some Astrologers, the latitude is also important in considering the aspects in order for them to be considered a real aspect. The latitude of a Planet is its northern or southern position (as opposed to its Longitude position which is determined by its position in the Zodiac). A Planet is normally only Northern or Southern a few degrees however when considering aspects there are a few rules that are important concerning latitude. For a conjunction, it is stated that the conjunction must be equal in longitude as well as latitude. This is especially the case when we speak of Cazimi, or a Planet in the Heart of the Sun. When they are equal latitude and longitude, they are seen as a true Cazimi aspect. Whereas if we find that the Planet shares not the same latitude with the Sun (which is always at 0°) then it is considered in combustion. Planets who are in opposition must be equal degrees in the other pole; for example, if we find Saturn 2°N, and Jupiter in 2°S, they shall be in opposition. Quartile or square has no latitude consideration. Trines must be half of the number of latitude, for example; Saturn 1°N, and Jupiter 30'S. Finally, sextile is half the number but on the same side, therefore Saturn in 2°N is in sextile with Jupiter who is 1°N.