Math can't be avoided if you're learning astrology, and whilst you don't need to be all that bright to get your head around the basics - you really CAN'T pull the 'I read charts from intuition' card. Whether we like it or not, astrology is a system, and that system has rules. It's also got its own language. An aspect is simply the word used to describe the distance between two planets in an astrological chart. To make life easier for us mere mortals, some wise soul decided to name the aspects individually, so we didn't have to refer to each aspect by the number of degrees of separation. By naming the aspect a conjunction, trine or square we can easily understand the nature of the relationship between the two points or planets because they show us how the planets are interacting with each other, which can be relatively easy (trine), or difficult (square).
Aspects in the chart can give a huge amount of information and often reinforce what is already shown in the signs and planets, and some astrologers will do away with signs and houses completely and rely on aspects alone for their interpretations. Hard aspects are never to be feared because it's the squares and oppositions which can help to develop character. What they represent are obstacles, blockages, and hurdles; they don't deny us achievement in life, and they will frequently offer a gift if you learn to handle them wisely.
If you start a discussion on acceptable orbs, you can pretty much guarantee that it'll end up in an argument. Personally, I think it's stupid to assume that orbs should be the same for transits as they are for psychological astrology. Transits are all about timing, so it makes sense to keep orbs tighter, whereas the personality is much more fluid. If personal planets are being aspected during a transit, then it makes sense that any planets connected to them in the natal chart will be brought into closer focus, and even a wide square will become more apparent. With this in mind; I use up to 10 degrees for the conjunction, opposition, trine, and square, up to 6 degrees for a sextile and only 2 degrees for the "minor" ones with the exception of the quincunx which I allow up to 5 degrees. If I'm working with fixed stars, asteroids or the transneptunian objects I only use a maximum of 2 degree orb whatever the aspect. I think we have to give the bulk of the weight to the main heavenly bodies and to use wide orbs for the asteroids is just plain ignorant. Their influence is secondary to planets and angles, and what tends to happen is astrologers get carried away using asteroids to explain the personality and life events when the basic natal chart already says the same thing. One point to remember with orbs is that we need to avoid an overlay of aspects; for example, the bi-quintile and quincunx are only 6 degrees apart, so it's important to look closely to see how the energy of the planets is playing out. If you're just starting out with astrology, try to stick to using the conjunction, trine, sextile, square and opposition aspects otherwise you'll just overload with information. Once you're comfortable with those, you can move on to study the quintile, quincunx, semi-sextile and semi-square. Aspects such as the decile, novile, septile and quindecile will probably need to be calculated by hand.
Conjunction 0° - When two or more planets sit side by side. Planets in conjunction will modify the expression of one another as it tends to lead to a kind blending rather than the planets operating individually.
Semi-sextile 30° - I find this a difficult aspect to integrate yet some say it can create a support system between the planets involved. There can be a little bit of understanding between the planets concerned, but a fair amount of effort is needed to get them to work together.
Semi-square 45° - This aspect can be moderately difficult and create tension but not to the degree that a square or opposition would. Some astrologers say it can indicate trouble in the making.
Sextile 60° - This aspect is considered easy but needs effort to manifest fully. It is said to show a certain amount of talent in handling the energies between the planets involved.
Quintile 72° - Said to indicate hidden creative gifts or a creative way of expressing the energies of the planets involved. Appears frequently in charts of artists or those who are a little bit unique.
Square 90° - This aspect is said to create tension, this isn't always a bad thing and can lead to a certain amount of strength being built within the character. Blockages and obstacles are indicated but so is perseverance through difficulties.
Trine 120° - indicates a smooth flow of energy between the planets involved. The only downside is a possibility of laziness due to the easy flow of energy.
Sesquiquadrate 135° - a difficult aspect even though signs involved may be of the same element. This aspect needs considerable effort to resolve any issues raised by the planets involved.
Bi-Quintile 144° - much the same as the quintile but there can be a strong sense of urgency to manifest the potentials shown. Some astrologers say the bi-quintile is the meeting of two gifts, or that it takes someone else having a planet in aspect to the bi-quintile to help manifest the potential.
Quincunx 150° - one of the most challenging aspects, though some say its effects are neutral, and it depends on the planets involved. A chart with more than two or three and the person is likely to be a very determined individual, especially if the Sun, Mars, Saturn, or Pluto are involved.
Opposition 180° - it literally represents an opposition, two forces pulling against one another. Planets in opposition both need to be honoured, but we often reject one in favour of the other.
Parallel - Acts similar to a conjunction but is weaker in strength. Use max 1.5 orb.
Contra-parallel - Acts similar to an opposition but is weaker in strength. Use max 1.5° orb. To find parallels or contraparallels you need to look at the declinations of the planets. There will be a number followed by the letter N or S, a parallel will have the same number and letter, whereas the contraparallel has the same number but different letter.
Decile 36° - Zane Stein says the decile is an aspect that gives abilities to support others in a unique way, and there could be technical ability, but gifts and talents may be wasted or unappreciated. He recommends some self-development. It's not an aspect I've worked with much, but I'll try to take more notice in the future, I can tell you that I've got Sun decile Pluto and what he says fits well for me.
Novile 40° - it is said this aspect can indicate some sort of bondage between the planets but that once freed; a new form of expression of the energies can emerge and be used for the greater good. I can say that there does indeed seem to be two levels for this aspect to operate at, and there could be an element of stigma attached to the individual because of their desire for independence in whatever manner is shown by the planets involved.
Septile 51° 25 mins 42.8 seconds - according to Zane Stein this offers the ability to see what others miss, the septile is the mystery element - that which is hidden from reality and this aspect can hold a strong feeling of destiny. Some say this aspect is a very spiritual one that indicates a degree of artistic expression, but I'm gonna reject that label and say it could be an indicator for low latent inhibition, especially if Mercury is involved.
Quindecile 165° - according to Rikki Reeves this aspect creates a degree of obsession which I've found to be an accurate description, the planets and signs involved will reveal more. I've got the Moon (Aries) quindecile Pluto (Libra), and I admit to getting obsessive about my need for emotional freedom as well as having the need to experience powerful feelings.
There are a couple of other points I want to mention here...
Septile: Because the Septile is basically the division of the circle by seven, some astrologers will look for other Septile aspects such as the bi-septile 102° 51'26, the tri-septile 154° 17'08. Septiles are said to have a spiritual feel and be linked to instances which appear to have a sense of divine intervention.
The Novile divides the circle by nine, so you can have the novile at 40°, the bi-novile 80°, the tri-novile is the trine 120°, the quad-novile at 160°. Noviles are said to be linked to being tested in some way. Others describe it as an initiation of sorts.
There is no bi-decile because it's known as the quintile, but then we have the tri-decile which is 108°. The tredecile will come up in conversations connected to genius of some sort.
Online Reading For the Minor Aspects
See here for an explanation of the tredecile, septile, quintile combination in play.
Alice Portman's site is useful for the noviles, quintiles, and septiles.
For an extensive study of the aspects I can recommend two books in particular:
Aspects and Personality by Karen Hamaker-Zondag
Aspects in Astrology by Sue Tompkins